The infallible declarations of Vatican 1 guarantee the validity of the Post V2 popes:
http://www.papalencyclicals.net/Councils/ecuм20.htm#SESSION%204%20:%2018%20July%201870
On the permanence of the primacy of blessed Peter in the Roman pontiffs That which our Lord Jesus Christ, the prince of shepherds and great shepherd of the sheep, established in the blessed apostle Peter, for the continual salvation and permanent benefit of the church, must of necessity remain for ever, by Christ's authority, in the church which, founded as it is upon a rock, will stand firm until the end of time .
For no one can be in doubt, indeed it was known in every age that the holy and most blessed Peter, prince and head of the apostles, the pillar of faith and the foundation of the catholic church, received the keys of the kingdom from our lord Jesus Christ, the saviour and redeemer of the human race, and that to this day and for ever he lives and presides and exercises judgment in his successors the bishops of the holy Roman see, which he founded and consecrated with his blood .
Therefore whoever succeeds to the chair of Peter obtains by the institution of Christ himself, the primacy of Peter over the whole church. So what the truth has ordained stands firm, and blessed Peter perseveres in the rock-like strength he was granted, and does not abandon that guidance of the church which he once received .
For this reason it has always been necessary for every church--that is to say the faithful throughout the world--to be in agreement with the Roman church because of its more effective leadership. In consequence of being joined, as members to head, with that see, from which the rights of sacred communion flow to all, they will grow together into the structure of a single body .
Therefore,
if anyone says that
it is not by the institution of Christ the lord himself (that is to say, by divine law) that blessed Peter should have perpetual successors in the primacy over the whole church; or that
the Roman pontiff is not the successor of blessed Peter in this primacy:
let him be anathema.
In order, then, that
the episcopal office should be one and undivided and that,
by the union of the clergy,
the whole multitude of believers should be held together in the unity of
faith and
communion,
he set blessed Peter over the rest of the apostles and
instituted in him the permanent principle of both unities and
their visible foundation.
Upon the strength of this foundation was to be built the eternal temple, and the church whose topmost part reaches heaven was to rise upon the firmness of this foundation .
And since the gates of hell trying, if they can, to overthrow the church, make their assault with a hatred that increases day by day against its divinely laid foundation,
we judge it necessary,
with the approbation of the sacred council, and
for the protection, defence and growth of the catholic flock,
to propound the doctrine concerning the
institution,
permanence and
nature
of the sacred and apostolic primacy, upon which the strength and coherence of the whole church depends.
This doctrine is to be believed and held by all the faithful in accordance with the ancient and unchanging faith of the whole church.Furthermore, we shall proscribe and condemn the contrary errors which are so harmful to the Lord's flock.
Thus, all who hold the contrary erroneous position are bound under pain of grievous mortal sin to renounce their error.