Theorem fcofo 5787

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 997	. 2 |- ( ( F : A --> B /\ S : B --> A /\ ( F o. S ) = ( _I |` B ) ) -> F : A --> B )
2		ffvelcdm 5651	. . . . 5 |- ( ( S : B --> A /\ y e. B ) -> ( S ` y ) e. A )
3	2	3ad2antl2 1160	. . . 4 |- ( ( ( F : A --> B /\ S : B --> A /\ ( F o. S ) = ( _I |` B ) ) /\ y e. B ) -> ( S ` y ) e. A )
4		simpl3 1002	. . . . . 6 |- ( ( ( F : A --> B /\ S : B --> A /\ ( F o. S ) = ( _I |` B ) ) /\ y e. B ) -> ( F o. S ) = ( _I |` B ) )
5	4	fveq1d 5519	. . . . 5 |- ( ( ( F : A --> B /\ S : B --> A /\ ( F o. S ) = ( _I |` B ) ) /\ y e. B ) -> ( ( F o. S ) ` y ) = ( ( _I |` B ) ` y ) )
6		fvco3 5589	. . . . . 6 |- ( ( S : B --> A /\ y e. B ) -> ( ( F o. S ) ` y ) = ( F ` ( S ` y ) ) )
7	6	3ad2antl2 1160	. . . . 5 |- ( ( ( F : A --> B /\ S : B --> A /\ ( F o. S ) = ( _I |` B ) ) /\ y e. B ) -> ( ( F o. S ) ` y ) = ( F ` ( S ` y ) ) )
8		fvresi 5711	. . . . . 6 |- ( y e. B -> ( ( _I |` B ) ` y ) = y )
9	8	adantl 277	. . . . 5 |- ( ( ( F : A --> B /\ S : B --> A /\ ( F o. S ) = ( _I |` B ) ) /\ y e. B ) -> ( ( _I |` B ) ` y ) = y )
10	5, 7, 9	3eqtr3rd 2219	. . . 4 |- ( ( ( F : A --> B /\ S : B --> A /\ ( F o. S ) = ( _I |` B ) ) /\ y e. B ) -> y = ( F ` ( S ` y ) ) )
11		fveq2 5517	. . . . . 6 |- ( x = ( S ` y ) -> ( F ` x ) = ( F ` ( S ` y ) ) )
12	11	eqeq2d 2189	. . . . 5 |- ( x = ( S ` y ) -> ( y = ( F ` x ) <-> y = ( F ` ( S ` y ) ) ) )
13	12	rspcev 2843	. . . 4 |- ( ( ( S ` y ) e. A /\ y = ( F ` ( S ` y ) ) ) -> E. x e. A y = ( F ` x ) )
14	3, 10, 13	syl2anc 411	. . 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 2550	. 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 5665	. 2 |- ( F : A -onto-> B <-> ( F : A --> B /\ A. y e. B E. x e. A y = ( F ` x ) ) )
17	1, 15, 16	sylanbrc 417	1 |- ( ( F : A --> B /\ S : B --> A /\ ( F o. S ) = ( _I |` B ) ) -> F : A -onto-> B )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 978   = wceq 1353   e. wcel 2148   A.wral 2455   E.wrex 2456   _I cid 4290   |` cres 4630   o. ccom 4632   -->wf 5214   -onto->wfo 5216   ` cfv 5218

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-fo 5224  df-fv 5226

This theorem is referenced by:  fcof1o  5792