You will not find a verse that says "this is about the Bible". So you have to "connect the dots".
Peter said We have also a more sure word of prophecy; whereunto ye do well that ye take heed, as unto a light that shineth in a dark place, until the day dawn, and the day star arise in your hearts: Knowing this first, that no prophecy of the scripture is of any private interpretation. For the prophecy came not in old time by the will of man: but holy men of God spake as they were moved by the Holy Ghost. (2 Peter 1:19-21).
Please note the following:
1. "We have a more sure word of prophecy."
This is in relation to the written Word of God. Peter puts his own, and all the apostles writings into the same category as "prophecy of the Scripture", and tells us that absolutely all the writers of the OT (called "Scripture" here) wrote under divine inspiration ("moved by the Holy Ghost").
2. Then Peter equates ALL of Paul's epistles with the OT Scriptures:
And account that the longsuffering of our Lord is salvation; even as our beloved brother Paul also according to the wisdom given unto him hath written unto you; As also in all his epistles, speaking in them of these things; in which are some things hard to be understood, which they that are unlearned and unstable wrest, as they do also the other scriptures, unto their own destruction. (2 Peter 3:15,16)
3. Then Paul (before Peter) said this:
All scripture is given by inspiration of God, and is profitable for doctrine, for reproof, for correction, for instruction in righteousness: (2 Tim 3:16). Which implies a complete Bible.
4. Then John said this in the last chapter of the last book of the NT:
For I testify unto every man that heareth the words of the prophecy of this book, If any man shall add unto these things, God shall add unto him the plagues that are written in this book: (Rev 22:18) Was John speaking solely about Revelation, or was he applying this to the whole Bible, since he knew that all the apostles had passed on and he was the last one? the Bible closed with Revelation around 95-96 AD. Was it complete? Absolutely.
So did Paul's prophecy regarding the cessation of all prophecies come to pass? Absolutely. Therefore Paul was careful to say:
But when that which is perfect is come, then that which is in part shall be done away (1 Cor 13:10). Notice that he did not say "He who is perfect" but "that which is perfect". And the incomplete Bible ("that which is in part") was replaced with a complete Bible ("that which is complete")
Could that verse have also be stated as "that which is complete"? Absolutely.
Stephanus Textus Receptus 1550
οταν δε ελθη το τελειον τοτε το εκ μερους καταργηθησεται
Strong's Concordance
teleios: having reached its end, i.e. complete, by ext. perfect
Original Word: τέλειος, α, ον
Part of Speech: Adjective
Transliteration: teleios
Phonetic Spelling: (tel'-i-os)
Definition: having reached its end, complete, perfect
Usage: perfect, (a) complete in all its parts, (b) full grown, of full age, (c) specially of the completeness of Christian character.
The verses below verse 10 do not relate to the Bible, but to the perfection of the Christian. When we see God and Christ face-to-face after the Resurrection/Rapture, we all will be perfect and glorified.
For now we see through a glass, darkly; but then face to face: now I know in part; but then shall I know even as also I am known. (1 Cor 13:12)