Theorem foco2 6379

Description: If a composition of two functions is surjective, then the function on the left is surjective. (Contributed by Jeff Madsen, 16-Jun-2011.) (Proof shortened by JJ, 14-Jul-2021.)

Assertion

Ref	Expression
foco2	|- ( ( F : B --> C /\ G : A --> B /\ ( F o. G ) : A -onto-> C ) -> F : B -onto-> C )

Proof of Theorem foco2

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

Step	Hyp	Ref	Expression
1		foelrn 6378	. . . . . 6 |- ( ( ( F o. G ) : A -onto-> C /\ y e. C ) -> E. z e. A y = ( ( F o. G ) ` z ) )
2		ffvelrn 6357	. . . . . . . . 9 |- ( ( G : A --> B /\ z e. A ) -> ( G ` z ) e. B )
3		fvco3 6275	. . . . . . . . 9 |- ( ( G : A --> B /\ z e. A ) -> ( ( F o. G ) ` z ) = ( F ` ( G ` z ) ) )
4		fveq2 6191	. . . . . . . . . . 11 |- ( x = ( G ` z ) -> ( F ` x ) = ( F ` ( G ` z ) ) )
5	4	eqeq2d 2632	. . . . . . . . . 10 |- ( x = ( G ` z ) -> ( ( ( F o. G ) ` z ) = ( F ` x ) <-> ( ( F o. G ) ` z ) = ( F ` ( G ` z ) ) ) )
6	5	rspcev 3309	. . . . . . . . 9 |- ( ( ( G ` z ) e. B /\ ( ( F o. G ) ` z ) = ( F ` ( G ` z ) ) ) -> E. x e. B ( ( F o. G ) ` z ) = ( F ` x ) )
7	2, 3, 6	syl2anc 693	. . . . . . . 8 |- ( ( G : A --> B /\ z e. A ) -> E. x e. B ( ( F o. G ) ` z ) = ( F ` x ) )
8		eqeq1 2626	. . . . . . . . 9 |- ( y = ( ( F o. G ) ` z ) -> ( y = ( F ` x ) <-> ( ( F o. G ) ` z ) = ( F ` x ) ) )
9	8	rexbidv 3052	. . . . . . . 8 |- ( y = ( ( F o. G ) ` z ) -> ( E. x e. B y = ( F ` x ) <-> E. x e. B ( ( F o. G ) ` z ) = ( F ` x ) ) )
10	7, 9	syl5ibrcom 237	. . . . . . 7 |- ( ( G : A --> B /\ z e. A ) -> ( y = ( ( F o. G ) ` z ) -> E. x e. B y = ( F ` x ) ) )
11	10	rexlimdva 3031	. . . . . 6 |- ( G : A --> B -> ( E. z e. A y = ( ( F o. G ) ` z ) -> E. x e. B y = ( F ` x ) ) )
12	1, 11	syl5 34	. . . . 5 |- ( G : A --> B -> ( ( ( F o. G ) : A -onto-> C /\ y e. C ) -> E. x e. B y = ( F ` x ) ) )
13	12	impl 650	. . . 4 |- ( ( ( G : A --> B /\ ( F o. G ) : A -onto-> C ) /\ y e. C ) -> E. x e. B y = ( F ` x ) )
14	13	ralrimiva 2966	. . 3 |- ( ( G : A --> B /\ ( F o. G ) : A -onto-> C ) -> A. y e. C E. x e. B y = ( F ` x ) )
15	14	anim2i 593	. 2 |- ( ( F : B --> C /\ ( G : A --> B /\ ( F o. G ) : A -onto-> C ) ) -> ( F : B --> C /\ A. y e. C E. x e. B y = ( F ` x ) ) )
16		3anass 1042	. 2 |- ( ( F : B --> C /\ G : A --> B /\ ( F o. G ) : A -onto-> C ) <-> ( F : B --> C /\ ( G : A --> B /\ ( F o. G ) : A -onto-> C ) ) )
17		dffo3 6374	. 2 |- ( F : B -onto-> C <-> ( F : B --> C /\ A. y e. C E. x e. B y = ( F ` x ) ) )
18	15, 16, 17	3imtr4i 281	1 |- ( ( F : B --> C /\ G : A --> B /\ ( F o. G ) : A -onto-> C ) -> F : B -onto-> C )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   o. ccom 5118   -->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-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)