This is a carryover from Peter being chosen and directed by Jesus to "feed my sheep", plus Jesus' giving of his name that refers to "rock", and this was repeated by Ignatius of Antioch in a letter to Clement 1 at the end of the 1st century or beginning of the 2nd with the special designation as being the Bishop of Rome. But the more important thing along this line deals with the bishops as a group because of the collaboration between them, which did become increasingly more difficult due to distance and some dissention.
Yes, I'm familiar with the history and justification of the doctrine, but, of course, do not find the qualification duly substantiated. Basically, all the apostles were given the same authority and jurisdiction as Peter.
Matthew 18:15-19
15If your brother sins against you,
c go and confront him privately. If he listens to you, you have won your brother over.
16But if he will not listen, take one or two others along, so that ‘every matter may be established by the testimony of two or three witnesses.’
d 17If he refuses to listen to them, tell it to the church. And if he refuses to listen even to the church, regard him as you would a pagan or a tax collector.
18Truly I tell you, whatever you bind on earth will be bound in heaven, and whatever you loose on earth will be loosed in heaven.
19Again, I tell you truly that if two of you on the earth agree about anything you ask for, it will be done for you by My Father in heaven.
20For where two or three gather together in My name, there am I with them.”