Theorem cocanfo 33512

Description: Cancellation of a surjective function from the right side of a composition. (Contributed by Jeff Madsen, 1-Jun-2011.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)

Assertion

Ref	Expression
cocanfo	|- ( ( ( F : A -onto-> B /\ G Fn B /\ H Fn B ) /\ ( G o. F ) = ( H o. F ) ) -> G = H )

Proof of Theorem cocanfo

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplr 792	. . . . . 6 |- ( ( ( ( F : A -onto-> B /\ G Fn B /\ H Fn B ) /\ ( G o. F ) = ( H o. F ) ) /\ y e. A ) -> ( G o. F ) = ( H o. F ) )
2	1	fveq1d 6193	. . . . 5 |- ( ( ( ( F : A -onto-> B /\ G Fn B /\ H Fn B ) /\ ( G o. F ) = ( H o. F ) ) /\ y e. A ) -> ( ( G o. F ) ` y ) = ( ( H o. F ) ` y ) )
3		simpl1 1064	. . . . . . 7 |- ( ( ( F : A -onto-> B /\ G Fn B /\ H Fn B ) /\ ( G o. F ) = ( H o. F ) ) -> F : A -onto-> B )
4		fof 6115	. . . . . . 7 |- ( F : A -onto-> B -> F : A --> B )
5	3, 4	syl 17	. . . . . 6 |- ( ( ( F : A -onto-> B /\ G Fn B /\ H Fn B ) /\ ( G o. F ) = ( H o. F ) ) -> F : A --> B )
6		fvco3 6275	. . . . . 6 |- ( ( F : A --> B /\ y e. A ) -> ( ( G o. F ) ` y ) = ( G ` ( F ` y ) ) )
7	5, 6	sylan 488	. . . . 5 |- ( ( ( ( F : A -onto-> B /\ G Fn B /\ H Fn B ) /\ ( G o. F ) = ( H o. F ) ) /\ y e. A ) -> ( ( G o. F ) ` y ) = ( G ` ( F ` y ) ) )
8		fvco3 6275	. . . . . 6 |- ( ( F : A --> B /\ y e. A ) -> ( ( H o. F ) ` y ) = ( H ` ( F ` y ) ) )
9	5, 8	sylan 488	. . . . 5 |- ( ( ( ( F : A -onto-> B /\ G Fn B /\ H Fn B ) /\ ( G o. F ) = ( H o. F ) ) /\ y e. A ) -> ( ( H o. F ) ` y ) = ( H ` ( F ` y ) ) )
10	2, 7, 9	3eqtr3d 2664	. . . 4 |- ( ( ( ( F : A -onto-> B /\ G Fn B /\ H Fn B ) /\ ( G o. F ) = ( H o. F ) ) /\ y e. A ) -> ( G ` ( F ` y ) ) = ( H ` ( F ` y ) ) )
11	10	ralrimiva 2966	. . 3 |- ( ( ( F : A -onto-> B /\ G Fn B /\ H Fn B ) /\ ( G o. F ) = ( H o. F ) ) -> A. y e. A ( G ` ( F ` y ) ) = ( H ` ( F ` y ) ) )
12		fveq2 6191	. . . . . 6 |- ( ( F ` y ) = x -> ( G ` ( F ` y ) ) = ( G ` x ) )
13		fveq2 6191	. . . . . 6 |- ( ( F ` y ) = x -> ( H ` ( F ` y ) ) = ( H ` x ) )
14	12, 13	eqeq12d 2637	. . . . 5 |- ( ( F ` y ) = x -> ( ( G ` ( F ` y ) ) = ( H ` ( F ` y ) ) <-> ( G ` x ) = ( H ` x ) ) )
15	14	cbvfo 6544	. . . 4 |- ( F : A -onto-> B -> ( A. y e. A ( G ` ( F ` y ) ) = ( H ` ( F ` y ) ) <-> A. x e. B ( G ` x ) = ( H ` x ) ) )
16	3, 15	syl 17	. . 3 |- ( ( ( F : A -onto-> B /\ G Fn B /\ H Fn B ) /\ ( G o. F ) = ( H o. F ) ) -> ( A. y e. A ( G ` ( F ` y ) ) = ( H ` ( F ` y ) ) <-> A. x e. B ( G ` x ) = ( H ` x ) ) )
17	11, 16	mpbid 222	. 2 |- ( ( ( F : A -onto-> B /\ G Fn B /\ H Fn B ) /\ ( G o. F ) = ( H o. F ) ) -> A. x e. B ( G ` x ) = ( H ` x ) )
18		eqfnfv 6311	. . . 4 |- ( ( G Fn B /\ H Fn B ) -> ( G = H <-> A. x e. B ( G ` x ) = ( H ` x ) ) )
19	18	3adant1 1079	. . 3 |- ( ( F : A -onto-> B /\ G Fn B /\ H Fn B ) -> ( G = H <-> A. x e. B ( G ` x ) = ( H ` x ) ) )
20	19	adantr 481	. 2 |- ( ( ( F : A -onto-> B /\ G Fn B /\ H Fn B ) /\ ( G o. F ) = ( H o. F ) ) -> ( G = H <-> A. x e. B ( G ` x ) = ( H ` x ) ) )
21	17, 20	mpbird 247	1 |- ( ( ( F : A -onto-> B /\ G Fn B /\ H Fn B ) /\ ( G o. F ) = ( H o. F ) ) -> G = H )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   o. ccom 5118   Fn wfn 5883   -->wf 5884   -onto->wfo 5886   ` 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-fo 5894  df-fv 5896

This theorem is referenced by: (None)