From Christ and through St. Peter the priesthood is handed on in perpetuity
Although, therefore, dearly beloved, we be found both weak and slothful in fulfilling the duties of our office, because, whatever devoted and vigorous action we desire to do, we are hindered by the frailty of our very condition; yet having the unceasing propitiation of the Almighty and perpetual Priest, who being like us and yet equal with the Father, brought down His Godhead even to things human, and raised His Manhood even to things Divine, we worthily and piously rejoice over His dispensation, whereby, though He has delegated the care of His sheep to many shepherds, yet He has not Himself abandoned the guardianship of His beloved flock.
And from His overruling and eternal protection we have received the support of the Apostles' aid also, which assuredly does not cease from its operation: and the strength of the foundation, on which the whole superstructure of the Church is reared, is not weakened by the weight of the temple that rests upon it. For the solidity of that faith which was praised in the chief of the Apostles is perpetual: and as that remains which Peter believed in Christ, so that remains which Christ instituted in Peter.
For when, as has been read in the Gospel lesson, the Lord had asked the disciples whom they believed Him to be amid the various opinions that were held, and the blessed Peter had replied, saying, “You are the Christ, the Son of the living God,” the Lord says, “Blessed are you, Simon Bar-Jona, because flesh and flood has not revealed it to you, but My Father, which is in heaven. And I say to you, that you are Peter, and upon this rock will I build My church, and the gates of Hades shall not prevail against it. And I will give unto you the keys of the kingdom of heaven. And whatsoever you shall bind on earth, shall be bound in heaven; and whatsoever you shall loose on earth, shall be loosed also in heaven.”
Source: Sermons (New Advent)