Vatican I defined this never failing faith of Peter in exactly the same way it was understood by St Robert Bellarmine:
"'I have prayed for thee that thy faith not fail; and when you have converted, strengthen your brethren' (Luke 22:31). From this text, St Bernard in letter 90 to Pope Innocent deduced that the Roman Pontiff teaching ex cathedra cannot err; and before him the same was said by Pope Lucius I in letter I to the Bishops of Spain and France, by Pope Felix I in a letter to Benignus, Pope Mark in a letter to Athanasius, Leo I in sermon 3..., Leo IX in a letter to Peter Patriarch of Antioch, Agatho in a letter to the Emperor Constantine IV which was read at the Sixth Council (act 4 and again act 8) and approved by the whole Council, Pope Paschal II at the Roman Council..., Innocent III in the chapter Majores on Baptism and its effect... Therefore, if the Roman Pontiff cannot err when he is teaching ex cathedra, certainly his judgement must be followed... For we read Acts ch 15 that the Council said: 'It has seemed good to the Holy Ghost and to us'; such also now is the Pontiff's teaching ex cathedra, whom we showed is always directed by the Holy Ghost so that he cannot err." - St Robert Bellarmine, On the Word of God, Lib 3, Cap 5