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Theorem fcof1 6011
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.
StepHypRef Expression
1 simpl 444 . 2  |-  ( ( F : A --> B  /\  ( R  o.  F
)  =  (  _I  |`  A ) )  ->  F : A --> B )
2 simprr 734 . . . . . . . 8  |-  ( ( ( F : A --> B  /\  ( R  o.  F )  =  (  _I  |`  A )
)  /\  ( (
x  e.  A  /\  y  e.  A )  /\  ( F `  x
)  =  ( F `
 y ) ) )  ->  ( F `  x )  =  ( F `  y ) )
32fveq2d 5723 . . . . . . 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 731 . . . . . . . 8  |-  ( ( ( F : A --> B  /\  ( R  o.  F )  =  (  _I  |`  A )
)  /\  ( (
x  e.  A  /\  y  e.  A )  /\  ( F `  x
)  =  ( F `
 y ) ) )  ->  F : A
--> B )
5 simprll 739 . . . . . . . 8  |-  ( ( ( F : A --> B  /\  ( R  o.  F )  =  (  _I  |`  A )
)  /\  ( (
x  e.  A  /\  y  e.  A )  /\  ( F `  x
)  =  ( F `
 y ) ) )  ->  x  e.  A )
6 fvco3 5791 . . . . . . . 8  |-  ( ( F : A --> B  /\  x  e.  A )  ->  ( ( R  o.  F ) `  x
)  =  ( R `
 ( F `  x ) ) )
74, 5, 6syl2anc 643 . . . . . . 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 740 . . . . . . . 8  |-  ( ( ( F : A --> B  /\  ( R  o.  F )  =  (  _I  |`  A )
)  /\  ( (
x  e.  A  /\  y  e.  A )  /\  ( F `  x
)  =  ( F `
 y ) ) )  ->  y  e.  A )
9 fvco3 5791 . . . . . . . 8  |-  ( ( F : A --> B  /\  y  e.  A )  ->  ( ( R  o.  F ) `  y
)  =  ( R `
 ( F `  y ) ) )
104, 8, 9syl2anc 643 . . . . . . 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 ) ) )
113, 7, 103eqtr4d 2477 . . . . . 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 732 . . . . . . 7  |-  ( ( ( F : A --> B  /\  ( R  o.  F )  =  (  _I  |`  A )
)  /\  ( (
x  e.  A  /\  y  e.  A )  /\  ( F `  x
)  =  ( F `
 y ) ) )  ->  ( R  o.  F )  =  (  _I  |`  A )
)
1312fveq1d 5721 . . . . . 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 ) )
1412fveq1d 5721 . . . . . 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 ) )
1511, 13, 143eqtr3d 2475 . . . . 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 5915 . . . . . 6  |-  ( x  e.  A  ->  (
(  _I  |`  A ) `
 x )  =  x )
175, 16syl 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 5915 . . . . . 6  |-  ( y  e.  A  ->  (
(  _I  |`  A ) `
 y )  =  y )
198, 18syl 16 . . . . 5  |-  ( ( ( F : A --> B  /\  ( R  o.  F )  =  (  _I  |`  A )
)  /\  ( (
x  e.  A  /\  y  e.  A )  /\  ( F `  x
)  =  ( F `
 y ) ) )  ->  ( (  _I  |`  A ) `  y )  =  y )
2015, 17, 193eqtr3d 2475 . . . 4  |-  ( ( ( F : A --> B  /\  ( R  o.  F )  =  (  _I  |`  A )
)  /\  ( (
x  e.  A  /\  y  e.  A )  /\  ( F `  x
)  =  ( F `
 y ) ) )  ->  x  =  y )
2120expr 599 . . 3  |-  ( ( ( F : A --> B  /\  ( R  o.  F )  =  (  _I  |`  A )
)  /\  ( x  e.  A  /\  y  e.  A ) )  -> 
( ( F `  x )  =  ( F `  y )  ->  x  =  y ) )
2221ralrimivva 2790 . 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 5995 . 2  |-  ( F : A -1-1-> B  <->  ( F : A --> B  /\  A. x  e.  A  A. y  e.  A  (
( F `  x
)  =  ( F `
 y )  ->  x  =  y )
) )
241, 22, 23sylanbrc 646 1  |-  ( ( F : A --> B  /\  ( R  o.  F
)  =  (  _I  |`  A ) )  ->  F : A -1-1-> B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697    _I cid 4485    |` cres 4871    o. ccom 4873   -->wf 5441   -1-1->wf1 5442   ` cfv 5445
This theorem is referenced by:  fcof1o  6017
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fv 5453
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