Theorem fcofo 5678

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 981	. 2 |- ( ( F : A --> B /\ S : B --> A /\ ( F o. S ) = ( _I |` B ) ) -> F : A --> B )
2		ffvelrn 5546	. . . . 5 |- ( ( S : B --> A /\ y e. B ) -> ( S ` y ) e. A )
3	2	3ad2antl2 1144	. . . 4 |- ( ( ( F : A --> B /\ S : B --> A /\ ( F o. S ) = ( _I |` B ) ) /\ y e. B ) -> ( S ` y ) e. A )
4		simpl3 986	. . . . . 6 |- ( ( ( F : A --> B /\ S : B --> A /\ ( F o. S ) = ( _I |` B ) ) /\ y e. B ) -> ( F o. S ) = ( _I |` B ) )
5	4	fveq1d 5416	. . . . 5 |- ( ( ( F : A --> B /\ S : B --> A /\ ( F o. S ) = ( _I |` B ) ) /\ y e. B ) -> ( ( F o. S ) ` y ) = ( ( _I |` B ) ` y ) )
6		fvco3 5485	. . . . . 6 |- ( ( S : B --> A /\ y e. B ) -> ( ( F o. S ) ` y ) = ( F ` ( S ` y ) ) )
7	6	3ad2antl2 1144	. . . . 5 |- ( ( ( F : A --> B /\ S : B --> A /\ ( F o. S ) = ( _I |` B ) ) /\ y e. B ) -> ( ( F o. S ) ` y ) = ( F ` ( S ` y ) ) )
8		fvresi 5606	. . . . . 6 |- ( y e. B -> ( ( _I |` B ) ` y ) = y )
9	8	adantl 275	. . . . 5 |- ( ( ( F : A --> B /\ S : B --> A /\ ( F o. S ) = ( _I |` B ) ) /\ y e. B ) -> ( ( _I |` B ) ` y ) = y )
10	5, 7, 9	3eqtr3rd 2179	. . . 4 |- ( ( ( F : A --> B /\ S : B --> A /\ ( F o. S ) = ( _I |` B ) ) /\ y e. B ) -> y = ( F ` ( S ` y ) ) )
11		fveq2 5414	. . . . . 6 |- ( x = ( S ` y ) -> ( F ` x ) = ( F ` ( S ` y ) ) )
12	11	eqeq2d 2149	. . . . 5 |- ( x = ( S ` y ) -> ( y = ( F ` x ) <-> y = ( F ` ( S ` y ) ) ) )
13	12	rspcev 2784	. . . 4 |- ( ( ( S ` y ) e. A /\ y = ( F ` ( S ` y ) ) ) -> E. x e. A y = ( F ` x ) )
14	3, 10, 13	syl2anc 408	. . 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 2503	. 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 5560	. 2 |- ( F : A -onto-> B <-> ( F : A --> B /\ A. y e. B E. x e. A y = ( F ` x ) ) )
17	1, 15, 16	sylanbrc 413	1 |- ( ( F : A --> B /\ S : B --> A /\ ( F o. S ) = ( _I |` B ) ) -> F : A -onto-> B )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 962   = wceq 1331   e. wcel 1480   A.wral 2414   E.wrex 2415   _I cid 4205   |` cres 4536   o. ccom 4538   -->wf 5114   -onto->wfo 5116   ` cfv 5118

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-fo 5124  df-fv 5126

This theorem is referenced by:  fcof1o  5683