Theorem 2fvidinvd 6554

Description: Show that two functions are inverse to each other by applying them twice to each value of their domains. (Contributed by AV, 13-Dec-2019.)

Hypotheses

Ref	Expression
2fvcoidd.f	|- ( ph -> F : A --> B )
2fvcoidd.g	|- ( ph -> G : B --> A )
2fvcoidd.i	|- ( ph -> A. a e. A ( G ` ( F ` a ) ) = a )
2fvidf1od.i	|- ( ph -> A. b e. B ( F ` ( G ` b ) ) = b )

Assertion

Ref	Expression
2fvidinvd	|- ( ph -> `' F = G )

Distinct variable groups:   A, a   F, a   G, a   B, b   F, b   G, b

Allowed substitution hints:   ph( a, b)   A( b)   B( a)

Proof of Theorem 2fvidinvd

Step	Hyp	Ref	Expression
1		2fvcoidd.f	. 2 |- ( ph -> F : A --> B )
2		2fvcoidd.g	. 2 |- ( ph -> G : B --> A )
3		2fvcoidd.i	. . 3 |- ( ph -> A. a e. A ( G ` ( F ` a ) ) = a )
4	1, 2, 3	2fvcoidd 6552	. 2 |- ( ph -> ( G o. F ) = ( _I |` A ) )
5		2fvidf1od.i	. . 3 |- ( ph -> A. b e. B ( F ` ( G ` b ) ) = b )
6	2, 1, 5	2fvcoidd 6552	. 2 |- ( ph -> ( F o. G ) = ( _I |` B ) )
7	1, 2, 4, 6	2fcoidinvd 6550	1 |- ( ph -> `' F = G )

Syntax hints:   -> wi 4   = wceq 1483   A.wral 2912   `'ccnv 5113   -->wf 5884   ` cfv 5888

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-8 1992  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-pow 4843  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-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896

This theorem is referenced by:  m2cpminv  20565