Theorem fcof1 6103

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 457	. 2 |- ( ( F : A --> B /\ ( R o. F ) = ( _I |` A ) ) -> F : A --> B )
2		simprr 756	. . . . . . . 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 5806	. . . . . . 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 753	. . . . . . . 8 |- ( ( ( F : A --> B /\ ( R o. F ) = ( _I |` A ) ) /\ ( ( x e. A /\ y e. A ) /\ ( F ` x ) = ( F ` y ) ) ) -> F : A --> B )
5		simprll 761	. . . . . . . 8 |- ( ( ( F : A --> B /\ ( R o. F ) = ( _I |` A ) ) /\ ( ( x e. A /\ y e. A ) /\ ( F ` x ) = ( F ` y ) ) ) -> x e. A )
6		fvco3 5880	. . . . . . . 8 |- ( ( F : A --> B /\ x e. A ) -> ( ( R o. F ) ` x ) = ( R ` ( F ` x ) ) )
7	4, 5, 6	syl2anc 661	. . . . . . 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 762	. . . . . . . 8 |- ( ( ( F : A --> B /\ ( R o. F ) = ( _I |` A ) ) /\ ( ( x e. A /\ y e. A ) /\ ( F ` x ) = ( F ` y ) ) ) -> y e. A )
9		fvco3 5880	. . . . . . . 8 |- ( ( F : A --> B /\ y e. A ) -> ( ( R o. F ) ` y ) = ( R ` ( F ` y ) ) )
10	4, 8, 9	syl2anc 661	. . . . . . 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 2505	. . . . . 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 754	. . . . . . 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 5804	. . . . . 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 5804	. . . . . 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 2503	. . . . 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 6016	. . . . . 6 |- ( x e. A -> ( ( _I |` A ) ` x ) = x )
17	5, 16	syl 16	. . . . 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 6016	. . . . . 6 |- ( y e. A -> ( ( _I |` A ) ` y ) = y )
19	8, 18	syl 16	. . . . 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 2503	. . . 4 |- ( ( ( F : A --> B /\ ( R o. F ) = ( _I |` A ) ) /\ ( ( x e. A /\ y e. A ) /\ ( F ` x ) = ( F ` y ) ) ) -> x = y )
21	20	expr 615	. . 3 |- ( ( ( F : A --> B /\ ( R o. F ) = ( _I |` A ) ) /\ ( x e. A /\ y e. A ) ) -> ( ( F ` x ) = ( F ` y ) -> x = y ) )
22	21	ralrimivva 2914	. 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 6083	. 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 664	1 |- ( ( F : A --> B /\ ( R o. F ) = ( _I |` A ) ) -> F : A -1-1-> B )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1370   e. wcel 1758   A.wral 2799   _I cid 4742   |` cres 4953   o. ccom 4955   -->wf 5525   -1-1->wf1 5526   ` cfv 5529

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fv 5537

This theorem is referenced by:  fcof1o  6109