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Theorem fcof1 5797
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 443 . 2  |-  ( ( F : A --> B  /\  ( R  o.  F
)  =  (  _I  |`  A ) )  ->  F : A --> B )
2 simprr 733 . . . . . . . 8  |-  ( ( ( F : A --> B  /\  ( R  o.  F )  =  (  _I  |`  A )
)  /\  ( (
x  e.  A  /\  y  e.  A )  /\  ( F `  x
)  =  ( F `
 y ) ) )  ->  ( F `  x )  =  ( F `  y ) )
32fveq2d 5529 . . . . . . 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 730 . . . . . . . 8  |-  ( ( ( F : A --> B  /\  ( R  o.  F )  =  (  _I  |`  A )
)  /\  ( (
x  e.  A  /\  y  e.  A )  /\  ( F `  x
)  =  ( F `
 y ) ) )  ->  F : A
--> B )
5 simprll 738 . . . . . . . 8  |-  ( ( ( F : A --> B  /\  ( R  o.  F )  =  (  _I  |`  A )
)  /\  ( (
x  e.  A  /\  y  e.  A )  /\  ( F `  x
)  =  ( F `
 y ) ) )  ->  x  e.  A )
6 fvco3 5596 . . . . . . . 8  |-  ( ( F : A --> B  /\  x  e.  A )  ->  ( ( R  o.  F ) `  x
)  =  ( R `
 ( F `  x ) ) )
74, 5, 6syl2anc 642 . . . . . . 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 739 . . . . . . . 8  |-  ( ( ( F : A --> B  /\  ( R  o.  F )  =  (  _I  |`  A )
)  /\  ( (
x  e.  A  /\  y  e.  A )  /\  ( F `  x
)  =  ( F `
 y ) ) )  ->  y  e.  A )
9 fvco3 5596 . . . . . . . 8  |-  ( ( F : A --> B  /\  y  e.  A )  ->  ( ( R  o.  F ) `  y
)  =  ( R `
 ( F `  y ) ) )
104, 8, 9syl2anc 642 . . . . . . 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 2325 . . . . . 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 731 . . . . . . 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 5527 . . . . . 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 5527 . . . . . 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 2323 . . . . 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 5711 . . . . . 6  |-  ( x  e.  A  ->  (
(  _I  |`  A ) `
 x )  =  x )
175, 16syl 15 . . . . 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 5711 . . . . . 6  |-  ( y  e.  A  ->  (
(  _I  |`  A ) `
 y )  =  y )
198, 18syl 15 . . . . 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 2323 . . . 4  |-  ( ( ( F : A --> B  /\  ( R  o.  F )  =  (  _I  |`  A )
)  /\  ( (
x  e.  A  /\  y  e.  A )  /\  ( F `  x
)  =  ( F `
 y ) ) )  ->  x  =  y )
2120expr 598 . . 3  |-  ( ( ( F : A --> B  /\  ( R  o.  F )  =  (  _I  |`  A )
)  /\  ( x  e.  A  /\  y  e.  A ) )  -> 
( ( F `  x )  =  ( F `  y )  ->  x  =  y ) )
2221ralrimivva 2635 . 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 5783 . 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 645 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 358    = wceq 1623    e. wcel 1684   A.wral 2543    _I cid 4304    |` cres 4691    o. ccom 4693   -->wf 5251   -1-1->wf1 5252   ` cfv 5255
This theorem is referenced by:  fcof1o  5803  injrec2  25119
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fv 5263
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