Theorem foco2 5904

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

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		simp1 1024	. 2 |- ( ( F : B --> C /\ G : A --> B /\ ( F o. G ) : A -onto-> C ) -> F : B --> C )
2		foelrn 5903	. . . . . 6 |- ( ( ( F o. G ) : A -onto-> C /\ y e. C ) -> E. z e. A y = ( ( F o. G ) ` z ) )
3		ffvelcdm 5788	. . . . . . . . . 10 |- ( ( G : A --> B /\ z e. A ) -> ( G ` z ) e. B )
4	3	adantll 476	. . . . . . . . 9 |- ( ( ( F : B --> C /\ G : A --> B ) /\ z e. A ) -> ( G ` z ) e. B )
5		fvco3 5726	. . . . . . . . . 10 |- ( ( G : A --> B /\ z e. A ) -> ( ( F o. G ) ` z ) = ( F ` ( G ` z ) ) )
6	5	adantll 476	. . . . . . . . 9 |- ( ( ( F : B --> C /\ G : A --> B ) /\ z e. A ) -> ( ( F o. G ) ` z ) = ( F ` ( G ` z ) ) )
7		fveq2 5648	. . . . . . . . . . 11 |- ( x = ( G ` z ) -> ( F ` x ) = ( F ` ( G ` z ) ) )
8	7	eqeq2d 2243	. . . . . . . . . 10 |- ( x = ( G ` z ) -> ( ( ( F o. G ) ` z ) = ( F ` x ) <-> ( ( F o. G ) ` z ) = ( F ` ( G ` z ) ) ) )
9	8	rspcev 2911	. . . . . . . . 9 |- ( ( ( G ` z ) e. B /\ ( ( F o. G ) ` z ) = ( F ` ( G ` z ) ) ) -> E. x e. B ( ( F o. G ) ` z ) = ( F ` x ) )
10	4, 6, 9	syl2anc 411	. . . . . . . 8 |- ( ( ( F : B --> C /\ G : A --> B ) /\ z e. A ) -> E. x e. B ( ( F o. G ) ` z ) = ( F ` x ) )
11		eqeq1 2238	. . . . . . . . 9 |- ( y = ( ( F o. G ) ` z ) -> ( y = ( F ` x ) <-> ( ( F o. G ) ` z ) = ( F ` x ) ) )
12	11	rexbidv 2534	. . . . . . . 8 |- ( y = ( ( F o. G ) ` z ) -> ( E. x e. B y = ( F ` x ) <-> E. x e. B ( ( F o. G ) ` z ) = ( F ` x ) ) )
13	10, 12	syl5ibrcom 157	. . . . . . 7 |- ( ( ( F : B --> C /\ G : A --> B ) /\ z e. A ) -> ( y = ( ( F o. G ) ` z ) -> E. x e. B y = ( F ` x ) ) )
14	13	rexlimdva 2651	. . . . . 6 |- ( ( F : B --> C /\ G : A --> B ) -> ( E. z e. A y = ( ( F o. G ) ` z ) -> E. x e. B y = ( F ` x ) ) )
15	2, 14	syl5 32	. . . . 5 |- ( ( F : B --> C /\ G : A --> B ) -> ( ( ( F o. G ) : A -onto-> C /\ y e. C ) -> E. x e. B y = ( F ` x ) ) )
16	15	impl 380	. . . 4 |- ( ( ( ( F : B --> C /\ G : A --> B ) /\ ( F o. G ) : A -onto-> C ) /\ y e. C ) -> E. x e. B y = ( F ` x ) )
17	16	ralrimiva 2606	. . 3 |- ( ( ( F : B --> C /\ G : A --> B ) /\ ( F o. G ) : A -onto-> C ) -> A. y e. C E. x e. B y = ( F ` x ) )
18	17	3impa 1221	. 2 |- ( ( F : B --> C /\ G : A --> B /\ ( F o. G ) : A -onto-> C ) -> A. y e. C E. x e. B y = ( F ` x ) )
19		dffo3 5802	. 2 |- ( F : B -onto-> C <-> ( F : B --> C /\ A. y e. C E. x e. B y = ( F ` x ) ) )
20	1, 18, 19	sylanbrc 417	1 |- ( ( F : B --> C /\ G : A --> B /\ ( F o. G ) : A -onto-> C ) -> F : B -onto-> C )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   e. wcel 2202   A.wral 2511   E.wrex 2512   o. ccom 4735   -->wf 5329   -onto->wfo 5331   ` cfv 5333

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fo 5339  df-fv 5341

This theorem is referenced by: (None)