Theorem fcof1 5958

Description: An application is injective if a retraction exists. Proposition 8 of [BourbakiEns] p. E.II.18. (Contributed by FL, 11-Nov-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)

Assertion

Ref	Expression
fcof1	|- ( ( F : A --> B /\ ( R o. F ) = ( _I |` A ) ) -> F : A -1-1-> B )

Proof of Theorem fcof1

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

Step	Hyp	Ref	Expression
1		simpl 109	. 2 |- ( ( F : A --> B /\ ( R o. F ) = ( _I |` A ) ) -> F : A --> B )
2		simprr 533	. . . . . . . 8 |- ( ( ( F : A --> B /\ ( R o. F ) = ( _I |` A ) ) /\ ( ( x e. A /\ y e. A ) /\ ( F ` x ) = ( F ` y ) ) ) -> ( F ` x ) = ( F ` y ) )
3	2	fveq2d 5676	. . . . . . 7 |- ( ( ( F : A --> B /\ ( R o. F ) = ( _I |` A ) ) /\ ( ( x e. A /\ y e. A ) /\ ( F ` x ) = ( F ` y ) ) ) -> ( R ` ( F ` x ) ) = ( R ` ( F ` y ) ) )
4		simpll 527	. . . . . . . 8 |- ( ( ( F : A --> B /\ ( R o. F ) = ( _I |` A ) ) /\ ( ( x e. A /\ y e. A ) /\ ( F ` x ) = ( F ` y ) ) ) -> F : A --> B )
5		simprll 539	. . . . . . . 8 |- ( ( ( F : A --> B /\ ( R o. F ) = ( _I |` A ) ) /\ ( ( x e. A /\ y e. A ) /\ ( F ` x ) = ( F ` y ) ) ) -> x e. A )
6		fvco3 5750	. . . . . . . 8 |- ( ( F : A --> B /\ x e. A ) -> ( ( R o. F ) ` x ) = ( R ` ( F ` x ) ) )
7	4, 5, 6	syl2anc 411	. . . . . . 7 |- ( ( ( F : A --> B /\ ( R o. F ) = ( _I |` A ) ) /\ ( ( x e. A /\ y e. A ) /\ ( F ` x ) = ( F ` y ) ) ) -> ( ( R o. F ) ` x ) = ( R ` ( F ` x ) ) )
8		simprlr 540	. . . . . . . 8 |- ( ( ( F : A --> B /\ ( R o. F ) = ( _I |` A ) ) /\ ( ( x e. A /\ y e. A ) /\ ( F ` x ) = ( F ` y ) ) ) -> y e. A )
9		fvco3 5750	. . . . . . . 8 |- ( ( F : A --> B /\ y e. A ) -> ( ( R o. F ) ` y ) = ( R ` ( F ` y ) ) )
10	4, 8, 9	syl2anc 411	. . . . . . 7 |- ( ( ( F : A --> B /\ ( R o. F ) = ( _I |` A ) ) /\ ( ( x e. A /\ y e. A ) /\ ( F ` x ) = ( F ` y ) ) ) -> ( ( R o. F ) ` y ) = ( R ` ( F ` y ) ) )
11	3, 7, 10	3eqtr4d 2277	. . . . . 6 |- ( ( ( F : A --> B /\ ( R o. F ) = ( _I |` A ) ) /\ ( ( x e. A /\ y e. A ) /\ ( F ` x ) = ( F ` y ) ) ) -> ( ( R o. F ) ` x ) = ( ( R o. F ) ` y ) )
12		simplr 529	. . . . . . 7 |- ( ( ( F : A --> B /\ ( R o. F ) = ( _I |` A ) ) /\ ( ( x e. A /\ y e. A ) /\ ( F ` x ) = ( F ` y ) ) ) -> ( R o. F ) = ( _I |` A ) )
13	12	fveq1d 5674	. . . . . 6 |- ( ( ( F : A --> B /\ ( R o. F ) = ( _I |` A ) ) /\ ( ( x e. A /\ y e. A ) /\ ( F ` x ) = ( F ` y ) ) ) -> ( ( R o. F ) ` x ) = ( ( _I |` A ) ` x ) )
14	12	fveq1d 5674	. . . . . 6 |- ( ( ( F : A --> B /\ ( R o. F ) = ( _I |` A ) ) /\ ( ( x e. A /\ y e. A ) /\ ( F ` x ) = ( F ` y ) ) ) -> ( ( R o. F ) ` y ) = ( ( _I |` A ) ` y ) )
15	11, 13, 14	3eqtr3d 2275	. . . . 5 |- ( ( ( F : A --> B /\ ( R o. F ) = ( _I |` A ) ) /\ ( ( x e. A /\ y e. A ) /\ ( F ` x ) = ( F ` y ) ) ) -> ( ( _I |` A ) ` x ) = ( ( _I |` A ) ` y ) )
16		fvresi 5879	. . . . . 6 |- ( x e. A -> ( ( _I |` A ) ` x ) = x )
17	5, 16	syl 14	. . . . 5 |- ( ( ( F : A --> B /\ ( R o. F ) = ( _I |` A ) ) /\ ( ( x e. A /\ y e. A ) /\ ( F ` x ) = ( F ` y ) ) ) -> ( ( _I |` A ) ` x ) = x )
18		fvresi 5879	. . . . . 6 |- ( y e. A -> ( ( _I |` A ) ` y ) = y )
19	8, 18	syl 14	. . . . 5 |- ( ( ( F : A --> B /\ ( R o. F ) = ( _I |` A ) ) /\ ( ( x e. A /\ y e. A ) /\ ( F ` x ) = ( F ` y ) ) ) -> ( ( _I |` A ) ` y ) = y )
20	15, 17, 19	3eqtr3d 2275	. . . 4 |- ( ( ( F : A --> B /\ ( R o. F ) = ( _I |` A ) ) /\ ( ( x e. A /\ y e. A ) /\ ( F ` x ) = ( F ` y ) ) ) -> x = y )
21	20	expr 375	. . 3 |- ( ( ( F : A --> B /\ ( R o. F ) = ( _I |` A ) ) /\ ( x e. A /\ y e. A ) ) -> ( ( F ` x ) = ( F ` y ) -> x = y ) )
22	21	ralrimivva 2626	. 2 |- ( ( F : A --> B /\ ( R o. F ) = ( _I |` A ) ) -> A. x e. A A. y e. A ( ( F ` x ) = ( F ` y ) -> x = y ) )
23		dff13 5943	. 2 |- ( F : A -1-1-> B <-> ( F : A --> B /\ A. x e. A A. y e. A ( ( F ` x ) = ( F ` y ) -> x = y ) ) )
24	1, 22, 23	sylanbrc 417	1 |- ( ( F : A --> B /\ ( R o. F ) = ( _I |` A ) ) -> F : A -1-1-> B )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   A.wral 2522   _I cid 4411   |` cres 4753   o. ccom 4755   -->wf 5350   -1-1->wf1 5351   ` cfv 5354

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 4230  ax-pow 4289  ax-pr 4324

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 3045  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fv 5362

This theorem is referenced by:  fcof1o  5964