Catholic Church because the Faith is Universal

My Catholic Friend attempted to prove that the Catholic Church is the church founded by Christ by simply telling me that Apostle Paul mentioned in the book of Romans that the Faith of Roman Christian is universal.

However, I told him that it couldn't be used as proof because of some reasons:

1. The letter of Paul to the Romans is not for all people in Rome but for the members of the Church of Christ in Rome.
2. Apostle Paul is not an Apostle of the Roman Catholic Church, but an apostle in the Church of Christ.
3. The faith mentioned by Apostle Paul is not the faith of the Roman Catholic Church but of the Church of Christ.

And we should also not forget that Christ built only his own Church or the Church of Christ. He didn't build any churches, especially the Roman Catholic Church, except his own Church.