SILER, Circuit Judge.

Gerald Molnar instituted this 42 U.S.C. §§ 1983 and 1985 action against Amy Allen and her employer Care House ("Care House Defendants"), and Detective Janice Pokely and the City of Troy ("City Defendants") (collectively, "the Defendants"); he also asserted supplemental state-law claims against the Defendants as well as his ex-wife Renee Molnar. He filed the instant suit after he was acquitted in state court of sexually assaulting his daughter Elizabeth Molnar, alleging that the Defendants' involvement in his state-court prosecution violated his federal rights. The district court granted summary judgment for the Defendants. For the following reasons, we AFFIRM.

I. Background

A. Facts Giving Rise to the Underlying Criminal Case Against Molnar

Renee and Gerald Molnar have been divorced since 1998. Molnar claims that Renee began accusing him of sexually abusing their youngest child Elizabeth in 1999, after he expressed concern that Renee's brother Brian Koch, a convicted sex offender, would have access to their children. Molnar also claims that Renee proceeded to make numerous false allegations to the authorities that he had sexually and physically abused their children, and that the authorities found each of these accusations to be unsubstantiated.

On October 8, 2003, Renee informed Detective Pokely that her husband had sexually abused nine-year-old Elizabeth by putting his hand down her pants and touching her vulva. Detective Pokely then set up an interview at Care House, and on October 16, 2003, Amy Allen, a Care House worker, interviewed Elizabeth. Elizabeth's statements were consistent and confirmed her mother's allegations.

Later that day, Detective Pokely met with Molnar and his girlfriend Mary Ellen Urban to discuss Elizabeth's allegations. Molnar denied that he had molested his daughter. He told Pokely that Renee's prior allegations had been found to be unsubstantiated. He also claimed that Urban had been present any time he was with his children. He then informed Pokely that when he had expressed concern to Renee about her brother Koch, Renee threatened to retaliate by filing false claims of sexual abuse. Urban later confirmed Molnar's story in an affidavit.

Pokely reviewed all the information gathered, including several incident reports of lewd behavior that named Molnar as a suspect, and submitted her narrative report to the Oakland County Prosecutor's office ("Prosecutor"). Her report included a discussion of the past unsubstantiated accusations made against Molnar. She noted that Molnar lived with Urban, but did not discuss Molnar's claim that Koch was a convicted sex offender or that Urban had stated that Molnar at no time acted inappropriately. Subsequently, the Prosecutor decided to pursue a criminal complaint against Molnar, charging him with criminal sexual conduct in the second degree. At Molnar's subsequent preliminary hearing on December 23, 2003, the court found probable cause to prosecute Molnar based on Elizabeth's "very believable" testimony. Earlier, on October 24, 2003, Referee Michael Hand had temporarily suspended Molnar's visitation rights with his children based on a finding that there was probable cause of sexual abuse. After a jury trial in 2005, Molnar was acquitted.

B. Federal Suit

After his acquittal, Molnar filed an amended complaint in the United States District Court for the Eastern District of Michigan. He alleged, inter alia, that he was falsely arrested and maliciously prosecuted without probable cause in violation of the Fourth Amendment, and that he was denied procedural and substantive due process and equal protection. He also contended that the Defendants engaged in a conspiracy to violate his constitutional rights in contravention of 42 U.S.C. § 1985. Finally, he asserted a Monell failure-to-train claim against the City, a supplemental state-law claim of intentional infliction of emotional distress against all the Defendants, and supplemental state-law claims of defamation per se and false light against Renee.

The City Defendants and the Care House Defendants filed motions for summary judgment. The district court granted both motions. Molnar appealed and has moved this court to take judicial notice of documents not contained in the district court record.

II. Analysis

A. Motion to Take Judicial Notice

Before addressing the merits of this appeal, we must resolve Molnar's motion to take judicial notice and to expand the record. Molnar seeks to supplement the record with fourteen documents that, he argues, demonstrate that he "was not afforded an opportunity to conduct discovery" due to an alleged direction from the district court that the parties hold discovery in abeyance.

Rule 10 of the Federal Rules of Appellate Procedure does not allow supplementation of the record in this case. See S E Shipping Corp. v. Chesapeake Ohio Ry. Co., 678 F.2d 636, 641 (6th Cir. 1982) (stating that the purpose of the rule is to correct "omissions from or misstatements in the record for appeal," and "not to introduce new evidence in the court of appeals"). Beyond a conclusory statement to the district court that summary judgment would be inappropriate because "discovery [was] yet to commence," Molnar did not attempt to submit these documents or otherwise argue the lack-of-discovery issue before the district court. He could have easily attached these documents with his motion to the district court. That he neglected to do so does not justify supplementation under Rule 10(e).

We also decline to exercise our equitable powers to allow Molnar to supplement the record. See Inland Bulk Transfer Co. v. Cummins Engine Co., 332 F.3d 1007, 1012 (6th Cir. 2003) (declining to supplement under our equitable powers). Given the absence of material disputed facts in this case and because the Defendants are entitled to judgment as a matter of law, allowing supplementation in this case would be inefficient and unnecessary.

B. Summary Judgment

Turning to the merits of Molnar's appeal, "[w]e conduct de novo review of decisions granting summary judgment, drawing all reasonable inferences in favor of the nonmoving party." Johnson v. Karnes, 398 F.3d 868, 873 (6th Cir. 2005).

1. Premature Grant of Summary Judgment

Molnar first claims that the district court prematurely granted summary judgment in favor of the Defendants, an argument raised for the first time on appeal. His oblique references to discovery made before the district court — first, his blanket observation that "discovery [was] yet to commence" in his motion opposing the City Defendants' motion to dismiss; and second, his statement in his motion opposing the Care House Defendants' motion for summary judgment that "discovery will provide additional evidence" — are not equivalent to raising the issue of premature motions for summary judgment. Molnar could have easily submitted a Rule 56(f) affidavit or, at the very least, objected to the grant of summary judgment before the district court. He did neither, and we therefore deem his premature summary judgment claim waived. See Scottsdale Ins. Co. v. Flowers, 513 F.3d 546, 552 (6th Cir. 2008) ("[A]n argument not raised before the district court is waived on appeal to this Court.").

2. Section 1983

To survive summary judgment, Molnar must (1) demonstrate a genuine issue of material fact as to the deprivation of a federal right and (2) show that the deprivation was caused by a person acting under color of state law. Johnson, 398 F.3d at 873. Molnar alleges four deprivations of a federal right: (1) false arrest, (2) malicious prosecution, (3) violations of procedural and substantive due process, and (4) violation of equal protection. Because each of these alleged deprivations depend on Molnar's contention that there was no probable cause to prosecute him, the success or failure of his § 1983 claim rests on his ability to prove a lack of probable cause. The district court held that Molnar was collaterally estopped from re-litigating the issue of probable cause given the state court's ruling in the preliminary hearing that there was probable cause to prosecute him.

We apply state law to determine whether a state court's determination of probable cause at a preliminary examination prohibits re-litigation of the issue in a subsequent § 1983 action. Darrah v. City of Oak Park, 255 F.3d 301, 311 (6th Cir. 2001). "Under Michigan law, [collateral estoppel] applies when (1) there is identity of parties across the proceedings, (2) there was a valid, final judgment in the first proceeding, (3) the same issue was actually litigated and necessarily determined in the first proceeding, and (4) the party against whom the doctrine is asserted had a full and fair opportunity to litigate the issue in the earlier proceeding". Id. (citing People v. Gates, 434 Mich. 146, 452 N.W.2d 627, 630-31 (1990)). Only the "identity of the party" and the "same issue" requirements are at issue in the instant case.

The Care House Defendants and the City Defendants are the same party for purposes of collateral estoppel. "[O]ne of the `critical factors' in applying collateral estoppel is the determination of whether the respective litigants were parties or privy to a party to an action in which a valid judgment has been rendered." Gates, 452 N.W.2d at 630. Here, assuming, arguendo, that the Care House Defendants are state actors, they would necessarily act in privity with the City Defendants, with whom they are alleged to have conspired. Id. (holding that the state's Department of Social Services and a County Prosecutor were in privity and satisfied the "identity of the party" requirement). Thus, the "identity of the party" requirement is satisfied.

Molnar focuses his appeal on contesting the "same issue" requirement. He argues that because he disputes the legality of his arrest and the integrity of Pokely's investigation rather than the state court's finding of probable cause, collateral estoppel should not bar his claim. He cites Darrah, 255 F.3d at 301, and Buttino v. City of Hamtramck, 87 Fed.Appx. 499 (6th Cir. 2004), in support. In Darrah, we held that collateral estoppel did not bar a § 1983 plaintiffs challenge because, under the facts, there was no identity of the issues. Darrah, 255 F.3d at 311. We noted that the plaintiff was not re-litigating the identical issue decided in the preliminary hearing — the existence of probable cause — but was instead seeking to demonstrate that the officer knowingly misstated material facts in order to establish probable cause. Id. at 311 (emphasis added). Thus, in Darrah, we proceeded to review the merits of the plaintiffs malicious prosecution claim, a claim we ultimately found lacking. Id. at 311-13.

Molnar argues that Officer Pokely (1) fabricated probable cause and (2) failed to disclose exculpatory information. As evidence of these claims, he points to Officer Pokely's failure to disclose in her police report that his girlfriend corroborated his statements and that Renee's brother was a pedophile. However, what was excluded from Pokely's police report is simply not material to resolving the issue of whose statements the state court relied on to establish probable cause. At the preliminary hearing, the Prosecutor called one witness, Elizabeth. Molnar, in rebuttal, called Renee, Pokely, and Allen. Even accepting Molnar's allegation that Detective Pokely "knowingly supplied the magistrate with false information," Darrah, 255 F.3d at 311, the state court did not rely on her testimony to establish probable cause. Rather, it bound him over for trial based on Elizabeth's "very believable" testimony. Thus, Molnar's reliance on Darrah is misplaced, because the state court relied on the victim's testimony to establish probable cause.

Since the state court did not rely on Detective Pokely's statements to establish probable cause, Molnar is left to argue that probable cause did not exist to bind him over for trial. This same issue was previously litigated in the preliminary hearing, and thus collateral estoppel bars Molnar from re-litigating it here. Gates, 452 N.W.2d at 630-31 (Mich. 1990). Thus, he cannot demonstrate a deprivation of a federal right, and his § 1983 claim fails as a matter of law. See Buttino, 87 Fed. Appx. at 505 (applying collateral estoppel where the state court did not rely on an officer's testimony to establish probable cause). His failure-to-train claim against the City of Troy fails as well.

3. Section 1985

Molnar also alleges that the Defendants engaged in a conspiracy to violate his federal constitutional right in violation of § 1985. Because he is collaterally estopped from contesting probable cause, his § 1985 claim similarly fails as a matter of law. Simply stated, he cannot demonstrate the existence of a conspiracy to violate his federal rights if he cannot first demonstrate an underlying deprivation of a federal right.

4. Supplemental State-Law Claims

Molnar also brought several state-law claims, including intentional infliction of emotional distress. He alleges that Detective Pokely "turn[ed] a blind eye to all the exculpatory evidence in order to pin a heinous crime on Plaintiff — sexual molestation of his minor daughter." Given Molnar's inability to come forth with evidence beyond such conclusory allegations, we affirm the district court's ruling as to this issue.

III. Conclusion

Molnar's motion to supplement the record is DENIED. We AFFIRM the district court's grant of summary judgment for the Defendants.