Theorem fcof1oinvd 6548

Description: Show that a function is the inverse of a bijective function if their composition is the identity function. Formerly part of proof of fcof1o 6551. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by AV, 15-Dec-2019.)

Hypotheses

Ref	Expression
fcof1oinvd.f	|- ( ph -> F : A -1-1-onto-> B )
fcof1oinvd.g	|- ( ph -> G : B --> A )
fcof1oinvd.b	|- ( ph -> ( F o. G ) = ( _I |` B ) )

Assertion

Ref	Expression
fcof1oinvd	|- ( ph -> `' F = G )

Proof of Theorem fcof1oinvd

Step	Hyp	Ref	Expression
1		fcof1oinvd.b	. . 3 |- ( ph -> ( F o. G ) = ( _I |` B ) )
2	1	coeq2d 5284	. 2 |- ( ph -> ( `' F o. ( F o. G ) ) = ( `' F o. ( _I |` B ) ) )
3		coass 5654	. . 3 |- ( ( `' F o. F ) o. G ) = ( `' F o. ( F o. G ) )
4		fcof1oinvd.f	. . . . . 6 |- ( ph -> F : A -1-1-onto-> B )
5		f1ococnv1 6165	. . . . . 6 |- ( F : A -1-1-onto-> B -> ( `' F o. F ) = ( _I |` A ) )
6	4, 5	syl 17	. . . . 5 |- ( ph -> ( `' F o. F ) = ( _I |` A ) )
7	6	coeq1d 5283	. . . 4 |- ( ph -> ( ( `' F o. F ) o. G ) = ( ( _I |` A ) o. G ) )
8		fcof1oinvd.g	. . . . 5 |- ( ph -> G : B --> A )
9		fcoi2 6079	. . . . 5 |- ( G : B --> A -> ( ( _I |` A ) o. G ) = G )
10	8, 9	syl 17	. . . 4 |- ( ph -> ( ( _I |` A ) o. G ) = G )
11	7, 10	eqtrd 2656	. . 3 |- ( ph -> ( ( `' F o. F ) o. G ) = G )
12	3, 11	syl5eqr 2670	. 2 |- ( ph -> ( `' F o. ( F o. G ) ) = G )
13		f1ocnv 6149	. . . . 5 |- ( F : A -1-1-onto-> B -> `' F : B -1-1-onto-> A )
14	4, 13	syl 17	. . . 4 |- ( ph -> `' F : B -1-1-onto-> A )
15		f1of 6137	. . . 4 |- ( `' F : B -1-1-onto-> A -> `' F : B --> A )
16	14, 15	syl 17	. . 3 |- ( ph -> `' F : B --> A )
17		fcoi1 6078	. . 3 |- ( `' F : B --> A -> ( `' F o. ( _I |` B ) ) = `' F )
18	16, 17	syl 17	. 2 |- ( ph -> ( `' F o. ( _I |` B ) ) = `' F )
19	2, 12, 18	3eqtr3rd 2665	1 |- ( ph -> `' F = G )

Syntax hints:   -> wi 4   = wceq 1483   _I cid 5023   `'ccnv 5113   |` cres 5116   o. ccom 5118   -->wf 5884   -1-1-onto->wf1o 5887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895

This theorem is referenced by:  2fcoidinvd  6550