Yes. Before the NT was written, Jesus said to the apostles to receive the Holy Spirit. However, from Matt 16, they knew Peter was to be the head of the whole church, and so they took this for granted which is why it did not have to be mentioned anywhere in scripture when the apostles and prophets wrote it. Correct?
Because the Apostles understood that Jesus is the Head of His Body, the Church. He commissioned Peter with specific gifts and duties for the ministry that belonged to Him. Where God guides, He provides. The primacy of Peter is found all through the Scriptures.
Matthew, chapter 16
18: 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. 19: I will give you the keys of the kingdom of heaven, and whatever you bind on earth shall be bound in heaven, and whatever you loose on earth shall be loosed in heaven."
Luke, chapter 22 – Jesus tells Peter to strengthen his brethren
31: “Simon, Simon, behold, Satan demanded to have you, that he might sift you like wheat,
32: but I have prayed for you that your faith may not fail; and when you have turned again, strengthen your brethren.”
Jesus’ prayer is singular. He did not pray for all of them, just for Peter. Therefore, all those that wish to be preserved in faith better get in with Peter!
John, chapter 21 – Peter is given Christ’s flock to shepherd
17: He said to him the third time, “Simon, son of John, do you love me?” Peter was grieved because he said to him the third time, “Do you love me?” And he said to him, “Lord, you know everything; you know that I love you.” Jesus said to him, "Feed my sheep.
Mark, chapter 16 – Peter is mentioned separate from the others
7: But go, tell his disciples and Peter that he is going before you to Galilee; there you will see him, as he told you."
Luke, chapter 24 – risen Jesus appeared to Peter first
34: who said, “The Lord has risen indeed, and has appeared to Simon!”
Acts, chapter 1 – Peter headed the meeting to elect Matthias
13: and when they had entered, they went up to the upper room, where they were staying, Peter and John and James and Andrew, Philip and Thomas, Bartholomew and Matthew, James the son of Alphaeus and Simon the Zealot and Judas the son of James.
14: All these with one accord devoted themselves to prayer, together with the women and Mary the mother of Jesus, and with his brothers.
15: In those days Peter stood up among the brethren (the company of persons was in all about a hundred and twenty), and said,
16: "Brethren, the scripture had to be fulfilled, which the Holy Spirit spoke beforehand by the mouth of David, concerning Judas who was guide to those who arrested Jesus.
Acts, chapter 2 - Peter leads the Apostles
14: But Peter, standing with the eleven, lifted up his voice and addressed them, "Men of Judea and all who dwell in Jerusalem, let this be known to you, and give ear to my words…
37: Now when they heard this they were cut to the heart, and said to Peter and the rest of the apostles, “Brethren, what shall we do?”
38: And Peter said to them, “Repent, and be baptized every one of you in the name of Jesus Christ for the forgiveness of your sins; and you shall receive the gift of the Holy Spirit.
39: For the promise is to you and to your children and to all that are far off, every one whom the Lord our God calls to him.”
40: And he testified with many other words and exhorted them, saying, “Save yourselves from this crooked generation.”
41: So those who received his word were baptized, and there were added that day about three thousand souls.