Theorem fcofo 5963

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 1024	. 2 |- ( ( F : A --> B /\ S : B --> A /\ ( F o. S ) = ( _I |` B ) ) -> F : A --> B )
2		ffvelcdm 5815	. . . . 5 |- ( ( S : B --> A /\ y e. B ) -> ( S ` y ) e. A )
3	2	3ad2antl2 1187	. . . 4 |- ( ( ( F : A --> B /\ S : B --> A /\ ( F o. S ) = ( _I |` B ) ) /\ y e. B ) -> ( S ` y ) e. A )
4		simpl3 1029	. . . . . 6 |- ( ( ( F : A --> B /\ S : B --> A /\ ( F o. S ) = ( _I |` B ) ) /\ y e. B ) -> ( F o. S ) = ( _I |` B ) )
5	4	fveq1d 5677	. . . . 5 |- ( ( ( F : A --> B /\ S : B --> A /\ ( F o. S ) = ( _I |` B ) ) /\ y e. B ) -> ( ( F o. S ) ` y ) = ( ( _I |` B ) ` y ) )
6		fvco3 5753	. . . . . 6 |- ( ( S : B --> A /\ y e. B ) -> ( ( F o. S ) ` y ) = ( F ` ( S ` y ) ) )
7	6	3ad2antl2 1187	. . . . 5 |- ( ( ( F : A --> B /\ S : B --> A /\ ( F o. S ) = ( _I |` B ) ) /\ y e. B ) -> ( ( F o. S ) ` y ) = ( F ` ( S ` y ) ) )
8		fvresi 5882	. . . . . 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 2276	. . . 4 |- ( ( ( F : A --> B /\ S : B --> A /\ ( F o. S ) = ( _I |` B ) ) /\ y e. B ) -> y = ( F ` ( S ` y ) ) )
11		fveq2 5675	. . . . . 6 |- ( x = ( S ` y ) -> ( F ` x ) = ( F ` ( S ` y ) ) )
12	11	eqeq2d 2246	. . . . 5 |- ( x = ( S ` y ) -> ( y = ( F ` x ) <-> y = ( F ` ( S ` y ) ) ) )
13	12	rspcev 2923	. . . 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 2617	. 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 5829	. 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 1005   = wceq 1398   e. wcel 2205   A.wral 2522   E.wrex 2523   _I cid 4414   |` cres 4756   o. ccom 4758   -->wf 5353   -onto->wfo 5355   ` cfv 5357

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 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3046  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fo 5363  df-fv 5365

This theorem is referenced by:  fcof1o  5968