Prove to whom? You're talking about enforcing federal law. What law gives the responsibility to what federal agency for examining candidate qualifications and ensuring they are correct?
The Constitution is law. Twentieth Amendment, Section Three. The federal agency is "Congress". They are guilty of eligibility nullification.
American presidential elections are conducted by the 50 states and the District of Columbia. Each state plus DC has its own election laws and eligibility requirements. We do not, in effect, have a federal election, we have 50 state elections plus a federal district election.
The Federal Elections Commission has jurisdiction over campaign finance law.
Congress has no statutory authority to disqualify a candidate absent a state level disqualification.
The chief election officer in each state, usually the Secretary of State, has the ultimate authority to determine who is eligible to be on the ballot. In 2012 there were 50 ballot challenges to Obama heard in courts or before Elections Boards in 22 states. None of the ballot challenges were successful. Absent a state’s chief elections officer ruling or a successful court/elections board ballot challenge, Congress had no grounds to act. Those facts are amplified by the fact that none of the defeated candidates who also received Electoral votes: McCain, Palin, Romney or Ryan ever filed suit; nor did the Republican National Committee on their behalf.
Congress had no grounds to charge that the President-elect had failed to qualify.