Theorem fcofo 6192

Description: An application is surjective if a section exists. Proposition 8 of [BourbakiEns] p. E.II.18. (Contributed by FL, 17-Nov-2011.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)

Assertion

Ref	Expression
fcofo	|- ( ( F : A --> B /\ S : B --> A /\ ( F o. S ) = ( _I |` B ) ) -> F : A -onto-> B )

Proof of Theorem fcofo

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

Step	Hyp	Ref	Expression
1		simp1 996	. 2 |- ( ( F : A --> B /\ S : B --> A /\ ( F o. S ) = ( _I |` B ) ) -> F : A --> B )
2		ffvelrn 6030	. . . . 5 |- ( ( S : B --> A /\ y e. B ) -> ( S ` y ) e. A )
3	2	3ad2antl2 1159	. . . 4 |- ( ( ( F : A --> B /\ S : B --> A /\ ( F o. S ) = ( _I |` B ) ) /\ y e. B ) -> ( S ` y ) e. A )
4		simpl3 1001	. . . . . 6 |- ( ( ( F : A --> B /\ S : B --> A /\ ( F o. S ) = ( _I |` B ) ) /\ y e. B ) -> ( F o. S ) = ( _I |` B ) )
5	4	fveq1d 5874	. . . . 5 |- ( ( ( F : A --> B /\ S : B --> A /\ ( F o. S ) = ( _I |` B ) ) /\ y e. B ) -> ( ( F o. S ) ` y ) = ( ( _I |` B ) ` y ) )
6		fvco3 5950	. . . . . 6 |- ( ( S : B --> A /\ y e. B ) -> ( ( F o. S ) ` y ) = ( F ` ( S ` y ) ) )
7	6	3ad2antl2 1159	. . . . 5 |- ( ( ( F : A --> B /\ S : B --> A /\ ( F o. S ) = ( _I |` B ) ) /\ y e. B ) -> ( ( F o. S ) ` y ) = ( F ` ( S ` y ) ) )
8		fvresi 6098	. . . . . 6 |- ( y e. B -> ( ( _I |` B ) ` y ) = y )
9	8	adantl 466	. . . . 5 |- ( ( ( F : A --> B /\ S : B --> A /\ ( F o. S ) = ( _I |` B ) ) /\ y e. B ) -> ( ( _I |` B ) ` y ) = y )
10	5, 7, 9	3eqtr3rd 2507	. . . 4 |- ( ( ( F : A --> B /\ S : B --> A /\ ( F o. S ) = ( _I |` B ) ) /\ y e. B ) -> y = ( F ` ( S ` y ) ) )
11		fveq2 5872	. . . . . 6 |- ( x = ( S ` y ) -> ( F ` x ) = ( F ` ( S ` y ) ) )
12	11	eqeq2d 2471	. . . . 5 |- ( x = ( S ` y ) -> ( y = ( F ` x ) <-> y = ( F ` ( S ` y ) ) ) )
13	12	rspcev 3210	. . . 4 |- ( ( ( S ` y ) e. A /\ y = ( F ` ( S ` y ) ) ) -> E. x e. A y = ( F ` x ) )
14	3, 10, 13	syl2anc 661	. . 3 |- ( ( ( F : A --> B /\ S : B --> A /\ ( F o. S ) = ( _I |` B ) ) /\ y e. B ) -> E. x e. A y = ( F ` x ) )
15	14	ralrimiva 2871	. 2 |- ( ( F : A --> B /\ S : B --> A /\ ( F o. S ) = ( _I |` B ) ) -> A. y e. B E. x e. A y = ( F ` x ) )
16		dffo3 6047	. 2 |- ( F : A -onto-> B <-> ( F : A --> B /\ A. y e. B E. x e. A y = ( F ` x ) ) )
17	1, 15, 16	sylanbrc 664	1 |- ( ( F : A --> B /\ S : B --> A /\ ( F o. S ) = ( _I |` B ) ) -> F : A -onto-> B )

Syntax hints:   -> wi 4   /\ wa 369   /\ w3a 973   = wceq 1395   e. wcel 1819   A.wral 2807   E.wrex 2808   _I cid 4799   |` cres 5010   o. ccom 5012   -->wf 5590   -onto->wfo 5592   ` cfv 5594

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fo 5600  df-fv 5602

This theorem is referenced by:  fcof1od  6198