The Catholic doctrine of the papacy is present, in all its essential components, in the primacy of St. Peter among the apostles:

1. Unique Prerogatives Directly Applied to St. Peter from Jesus

Matthew 16:18 (RSV, as throughout) And I tell you, you are Peter, and on this rock I will build my church; and the powers of death shall not prevail against it.

The rock (Greek, petra) is St. Peter himself, according to the overwhelming consensus of biblical scholars and commentators. Peter was the foundation-stone of the Church, and was regarded as the administrator.

Matthew 16:19a I will give you the keys of the kingdom of heaven . . .”

The “power of the keys” had to do with discipline and administrative authority with regard to the faith, as in Isaiah 22:22 (cf. Is 9:6; Job 12:14; Rev 3:7), and entailed the use of censures, excommunication, penance, and legislative powers.

Matthew 16:19b . . . whatever you bind on earth shall be bound in heaven, and whatever you loose on earth shall be loosed in heaven.

“Binding” and “loosing” were rabbinical terms, meaning to “forbid” and “permit” with reference to the interpretation of the law, and secondarily to “condemn” or “acquit.” Thus, St. Peter was given the authority to determine the rules for doctrine and practice. Bishops also these powers (Mt 18:17-18; Jn 20:23), but Peter was the only apostle who received them singularly, by name (making him preeminent).

Peter was regarded by Jesus as the chief shepherd after Himself (Jn 10:11; 21:15-17): over the universal Church, even though others have a similar but subordinate role (Acts 20:28; Eph 4:11; 1 Pet 5:2).

Jesus uniquely prayed for him, that his “faith may not fail”, and exhorted him to “strengthen” his “brethren” (Lk 22:32), and  informed him that he had received divine knowledge by a special revelation (Mt 16:17).

Jesus uniquely associated Himself and Peter in the miracle of the tribute-money (Mt 17:24-27), and taught from Peter’s boat. A  miraculous catch of fish followed (Lk 5:1-11).

2. Profound Significance of St. Peter's Name

Peter’s name occurs first in all lists of apostles (Mt 10:2; Mk 3:16; Lk 6:14; Acts 1:13). Matthew even called him the “first” (10:2). He is almost without exception named first whenever he appears with anyone else. In one (only?) example to the contrary, Galatians 2:9, where he (“Cephas”) is listed after James and before John, he is clearly preeminent in the entire context (e.g., 1:18-19; 2:7-8). He alone among the apostles received a new name, Rock (Jn 1:42; Mt 16:18).

Both Paul (1 Cor 15:4-8) and the two disciples on the road to Emmaus (Lk 24:34) distinguished the Lord’s post-Resurrection appearances to Peter from those to other apostles. Peter is often spoken of as distinct among apostles (Mk 1:36; Lk 9:28, 32; Acts 2:37; 5:29; 1 Cor 9:5), and his name is always the first listed of the “inner circle” of the disciples (Peter, James and John – Mt 17:1; 26:37, 40; Mk 5:37; 14:37).

He's mentioned more often than all the other disciples put together: 191 times (162 as Peter or Simon Peter, 23 as Simon, and 6 as Cephas). John is next (48).

3. St. Peter's Unique Actions, and “Historical Firsts”

Peter first confessed Christ’s divinity (Mt 16:16), first set out for, and entered the empty tomb (Lk 24:12; Jn 20:6), and led the apostles in fishing (Jn 21:2-3, 11). He was often the central figure in dramatic gospel scenes such as Jesus' walking on the water (Mt 14:28-32; Lk 5:1 ff., Mk 10:28; Mt 17:24 ff.).

He alone cast himself into the sea to come to Jesus (Jn 21:7), worked the first Christian miracle (Acts 3:6-12), and even healed by his shadow (Acts 5:15).

Peter was the first person after Christ to raise the dead (Acts 9:40) and to preach Christian repentance and baptism (Acts 2:38); and led the first recorded mass baptism (Acts 2:41). He was the first traveling missionary, and first exercised  “visitation of the churches” (Acts 9:32-38, 43).

4. Preeminence of St. Peter Acknowledged by Others

Peter was regarded by both the Jews (Acts 4:1-13) and the common people (Acts 2:37-41; 5:15) – even an angel (Mk 16:7) – as the leader of Christianity. Cornelius was told by an angel to seek Peter for Christian instruction (Acts 10:1-6), and an angel delivered Peter from prison: Acts 12:1-17), after the whole Church had offered “earnest prayer” (Acts 12:5). Paul went to Jerusalem specifically to see Peter for fifteen days in the beginning of his ministry (Gal 1:18; cf. Gal 2:9).

5. Examples of St. Peter's Singular Christian Authority

Peter’s words were the most important in the upper room before Pentecost (Acts 1:15-22), and he was the first person to speak (and “preach the gospel”) after Pentecost (Acts 2:14-41). His proclamation contained a fully authoritative interpretation of Scripture, a doctrinal decision and a disciplinary decree concerning members of the “House of Israel” (2:36).

He took the lead in calling for a replacement for Judas (Acts 1:22), uttered the first anathema (Acts 5:2-11), and was the first to recognize and refute heresy, in Simon Magus (Acts 8:14-24). He was the first to receive the Gentiles and to baptize them (Acts 10:9-48) He instructed the other apostles on the catholicity (universality) of the Church (Acts 11:5-17).

Peter presided over the first council of Christianity, and laid down the principles accepted by it (Acts 15:7-11). He was often the spokesman for the apostles (Mk 8:29; Mt 18:21; Lk 9:5; 12:41; Jn 6:67 ff.).

Peter was the first to judge the authenticity of spiritual gifts (Acts 2:14-21), acted as the chief bishop of the Church (1 Pet 5:1), in exhorting all other bishops. He interpreted prophecy (2 Pet 1:16-21), corrected those who misused Paul’s writings (2 Pet 3:15-16), and wrote his first epistle from Rome as its bishop, and as the universal bishop (or, pope) of the early Church. “Babylon” (1 Pet 5:13) was regarded as a code word for Rome.