Theorem foco2 5876

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 1021	. 2 |- ( ( F : B --> C /\ G : A --> B /\ ( F o. G ) : A -onto-> C ) -> F : B --> C )
2		foelrn 5875	. . . . . 6 |- ( ( ( F o. G ) : A -onto-> C /\ y e. C ) -> E. z e. A y = ( ( F o. G ) ` z ) )
3		ffvelcdm 5767	. . . . . . . . . 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 5704	. . . . . . . . . 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 5626	. . . . . . . . . . 11 |- ( x = ( G ` z ) -> ( F ` x ) = ( F ` ( G ` z ) ) )
8	7	eqeq2d 2241	. . . . . . . . . 10 |- ( x = ( G ` z ) -> ( ( ( F o. G ) ` z ) = ( F ` x ) <-> ( ( F o. G ) ` z ) = ( F ` ( G ` z ) ) ) )
9	8	rspcev 2907	. . . . . . . . 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 2236	. . . . . . . . 9 |- ( y = ( ( F o. G ) ` z ) -> ( y = ( F ` x ) <-> ( ( F o. G ) ` z ) = ( F ` x ) ) )
12	11	rexbidv 2531	. . . . . . . 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 2648	. . . . . 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 2603	. . 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 1218	. 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 5781	. 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 1002   = wceq 1395   e. wcel 2200   A.wral 2508   E.wrex 2509   o. ccom 4722   -->wf 5313   -onto->wfo 5315   ` cfv 5317

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-fo 5323  df-fv 5325

This theorem is referenced by: (None)