Theorem fcofo 5763

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 992	. 2 |- ( ( F : A --> B /\ S : B --> A /\ ( F o. S ) = ( _I |` B ) ) -> F : A --> B )
2		ffvelrn 5629	. . . . 5 |- ( ( S : B --> A /\ y e. B ) -> ( S ` y ) e. A )
3	2	3ad2antl2 1155	. . . 4 |- ( ( ( F : A --> B /\ S : B --> A /\ ( F o. S ) = ( _I |` B ) ) /\ y e. B ) -> ( S ` y ) e. A )
4		simpl3 997	. . . . . 6 |- ( ( ( F : A --> B /\ S : B --> A /\ ( F o. S ) = ( _I |` B ) ) /\ y e. B ) -> ( F o. S ) = ( _I |` B ) )
5	4	fveq1d 5498	. . . . 5 |- ( ( ( F : A --> B /\ S : B --> A /\ ( F o. S ) = ( _I |` B ) ) /\ y e. B ) -> ( ( F o. S ) ` y ) = ( ( _I |` B ) ` y ) )
6		fvco3 5567	. . . . . 6 |- ( ( S : B --> A /\ y e. B ) -> ( ( F o. S ) ` y ) = ( F ` ( S ` y ) ) )
7	6	3ad2antl2 1155	. . . . 5 |- ( ( ( F : A --> B /\ S : B --> A /\ ( F o. S ) = ( _I |` B ) ) /\ y e. B ) -> ( ( F o. S ) ` y ) = ( F ` ( S ` y ) ) )
8		fvresi 5689	. . . . . 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 2212	. . . 4 |- ( ( ( F : A --> B /\ S : B --> A /\ ( F o. S ) = ( _I |` B ) ) /\ y e. B ) -> y = ( F ` ( S ` y ) ) )
11		fveq2 5496	. . . . . 6 |- ( x = ( S ` y ) -> ( F ` x ) = ( F ` ( S ` y ) ) )
12	11	eqeq2d 2182	. . . . 5 |- ( x = ( S ` y ) -> ( y = ( F ` x ) <-> y = ( F ` ( S ` y ) ) ) )
13	12	rspcev 2834	. . . 4 |- ( ( ( S ` y ) e. A /\ y = ( F ` ( S ` y ) ) ) -> E. x e. A y = ( F ` x ) )
14	3, 10, 13	syl2anc 409	. . 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 2543	. 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 5643	. 2 |- ( F : A -onto-> B <-> ( F : A --> B /\ A. y e. B E. x e. A y = ( F ` x ) ) )
17	1, 15, 16	sylanbrc 415	1 |- ( ( F : A --> B /\ S : B --> A /\ ( F o. S ) = ( _I |` B ) ) -> F : A -onto-> B )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 973   = wceq 1348   e. wcel 2141   A.wral 2448   E.wrex 2449   _I cid 4273   |` cres 4613   o. ccom 4615   -->wf 5194   -onto->wfo 5196   ` cfv 5198

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fo 5204  df-fv 5206

This theorem is referenced by:  fcof1o  5768