I would recommend a thorough study of the early popes’ letters, especially those of Leo the Great, who is a saint to the Orthodox as well.
Papal authority can be proven from acts of jurisdiction done by the popes which presuppose universal jurisdiction-- particularly acts of jurisdiction over another patriarch.
From the ancient council collections, reproduced in Latin and Greek by Archbishop Mansi and in ACO, I would offer as an example the deposition of the heresiarch Dioscorus, patriarch of Alexandria, at the Council of Chalcedon. That was done not by Pope Leo in person, but by his legates-- a fortiori an act of authority: legates pronounce sentence on a patriarch. After enumerating the crimes of Dioscorus, which were very serious, Rome’s legates said:
Wherefore the most holy Leo, archbishop of great and old Rome, through us, and through this present most holy synod, together with the thrice-blessed and most glorious Apostle Peter, Rock and foundation of the Catholic Church, and the foundation of the orthodox faith, has stripped him (Dioscorus) of the episcopate and made him alien to the priestly dignity.