Theorem f1ocan1fv 33521

Description: Cancel a composition by a bijection by preapplying the converse. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)

Assertion

Ref	Expression
f1ocan1fv	|- ( ( Fun F /\ G : A -1-1-onto-> B /\ X e. B ) -> ( ( F o. G ) ` ( `' G ` X ) ) = ( F ` X ) )

Proof of Theorem f1ocan1fv

Step	Hyp	Ref	Expression
1		f1of 6137	. . . 4 |- ( G : A -1-1-onto-> B -> G : A --> B )
2	1	3ad2ant2 1083	. . 3 |- ( ( Fun F /\ G : A -1-1-onto-> B /\ X e. B ) -> G : A --> B )
3		f1ocnv 6149	. . . . . 6 |- ( G : A -1-1-onto-> B -> `' G : B -1-1-onto-> A )
4		f1of 6137	. . . . . 6 |- ( `' G : B -1-1-onto-> A -> `' G : B --> A )
5	3, 4	syl 17	. . . . 5 |- ( G : A -1-1-onto-> B -> `' G : B --> A )
6	5	3ad2ant2 1083	. . . 4 |- ( ( Fun F /\ G : A -1-1-onto-> B /\ X e. B ) -> `' G : B --> A )
7		simp3 1063	. . . 4 |- ( ( Fun F /\ G : A -1-1-onto-> B /\ X e. B ) -> X e. B )
8	6, 7	ffvelrnd 6360	. . 3 |- ( ( Fun F /\ G : A -1-1-onto-> B /\ X e. B ) -> ( `' G ` X ) e. A )
9		fvco3 6275	. . 3 |- ( ( G : A --> B /\ ( `' G ` X ) e. A ) -> ( ( F o. G ) ` ( `' G ` X ) ) = ( F ` ( G ` ( `' G ` X ) ) ) )
10	2, 8, 9	syl2anc 693	. 2 |- ( ( Fun F /\ G : A -1-1-onto-> B /\ X e. B ) -> ( ( F o. G ) ` ( `' G ` X ) ) = ( F ` ( G ` ( `' G ` X ) ) ) )
11		f1ocnvfv2 6533	. . . 4 |- ( ( G : A -1-1-onto-> B /\ X e. B ) -> ( G ` ( `' G ` X ) ) = X )
12	11	3adant1 1079	. . 3 |- ( ( Fun F /\ G : A -1-1-onto-> B /\ X e. B ) -> ( G ` ( `' G ` X ) ) = X )
13	12	fveq2d 6195	. 2 |- ( ( Fun F /\ G : A -1-1-onto-> B /\ X e. B ) -> ( F ` ( G ` ( `' G ` X ) ) ) = ( F ` X ) )
14	10, 13	eqtrd 2656	1 |- ( ( Fun F /\ G : A -1-1-onto-> B /\ X e. B ) -> ( ( F o. G ) ` ( `' G ` X ) ) = ( F ` X ) )

Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   e. wcel 1990   `'ccnv 5113   o. ccom 5118   Fun wfun 5882   -->wf 5884   -1-1-onto->wf1o 5887   ` 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-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-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:  f1ocan2fv  33522