How do we know the successors to the apostles had the same authority they did?

This is the real issue that separates Catholics from other Christians. I doubt very much that anybody, Fundamentalist or mainline Protestant, denies the authority of the apostles. The very fact that there were apostles and that Christ explicitly commanded them to “go forth and teach whatsoever I have commanded you” is too powerful to ignore or deny. The real problem is in the question of their transferring that mission to others who decidedly were not apostles.

That this teaching authority can reside elsewhere than solely among the apostles, is amply demonstrated. Of the four Gospels, two were written by men who were not apostles, and yet their writings bear the stamp of divine inspiration and authentic teaching. There is no record anywhere of Luke’s ever having met Christ, and he was certainly not one of the Twelve. Mark may have seen Christ in the flesh and indeed may have been the young man in the garden who fled naked when someone tore from his back the sheet he’d wrapped himself in. Matthias was chosen as apostle by the other apostles to replace Judas Iscariot. This choice was, according to the Acts of the Apostles, made directly under the inspiration of the Holy Spirit. Once elected, he was treated exactly as one of the original Twelve.

In other words, the Bible itself shows that Jesus gave the apostles the power to appoint successors.