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Theorem fcof1 5758
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
StepHypRef Expression
1 simpl 445 . 2  |-  ( ( F : A --> B  /\  ( R  o.  F
)  =  (  _I  |`  A ) )  ->  F : A --> B )
2 simprr 736 . . . . . . . 8  |-  ( ( ( F : A --> B  /\  ( R  o.  F )  =  (  _I  |`  A )
)  /\  ( (
x  e.  A  /\  y  e.  A )  /\  ( F `  x
)  =  ( F `
 y ) ) )  ->  ( F `  x )  =  ( F `  y ) )
32fveq2d 5489 . . . . . . 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 733 . . . . . . . 8  |-  ( ( ( F : A --> B  /\  ( R  o.  F )  =  (  _I  |`  A )
)  /\  ( (
x  e.  A  /\  y  e.  A )  /\  ( F `  x
)  =  ( F `
 y ) ) )  ->  F : A
--> B )
5 simprll 741 . . . . . . . 8  |-  ( ( ( F : A --> B  /\  ( R  o.  F )  =  (  _I  |`  A )
)  /\  ( (
x  e.  A  /\  y  e.  A )  /\  ( F `  x
)  =  ( F `
 y ) ) )  ->  x  e.  A )
6 fvco3 5557 . . . . . . . 8  |-  ( ( F : A --> B  /\  x  e.  A )  ->  ( ( R  o.  F ) `  x
)  =  ( R `
 ( F `  x ) ) )
74, 5, 6syl2anc 645 . . . . . . 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 742 . . . . . . . 8  |-  ( ( ( F : A --> B  /\  ( R  o.  F )  =  (  _I  |`  A )
)  /\  ( (
x  e.  A  /\  y  e.  A )  /\  ( F `  x
)  =  ( F `
 y ) ) )  ->  y  e.  A )
9 fvco3 5557 . . . . . . . 8  |-  ( ( F : A --> B  /\  y  e.  A )  ->  ( ( R  o.  F ) `  y
)  =  ( R `
 ( F `  y ) ) )
104, 8, 9syl2anc 645 . . . . . . 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 2326 . . . . . 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 734 . . . . . . 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 5487 . . . . . 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 5487 . . . . . 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 2324 . . . . 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 5672 . . . . . 6  |-  ( x  e.  A  ->  (
(  _I  |`  A ) `
 x )  =  x )
175, 16syl 17 . . . . 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 5672 . . . . . 6  |-  ( y  e.  A  ->  (
(  _I  |`  A ) `
 y )  =  y )
198, 18syl 17 . . . . 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 2324 . . . 4  |-  ( ( ( F : A --> B  /\  ( R  o.  F )  =  (  _I  |`  A )
)  /\  ( (
x  e.  A  /\  y  e.  A )  /\  ( F `  x
)  =  ( F `
 y ) ) )  ->  x  =  y )
2120expr 601 . . 3  |-  ( ( ( F : A --> B  /\  ( R  o.  F )  =  (  _I  |`  A )
)  /\  ( x  e.  A  /\  y  e.  A ) )  -> 
( ( F `  x )  =  ( F `  y )  ->  x  =  y ) )
2221ralrimivva 2636 . 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 5744 . 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 648 1  |-  ( ( F : A --> B  /\  ( R  o.  F
)  =  (  _I  |`  A ) )  ->  F : A -1-1-> B )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1628    e. wcel 1688   A.wral 2544    _I cid 4303    |` cres 4690    o. ccom 4692   -->wf 5217   -1-1->wf1 5218   ` cfv 5221
This theorem is referenced by:  fcof1o  5764  injrec2  24518
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fv 5229
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