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Theorem fcof1 6189
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 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 ) )
32fveq2d 5876 . . . . . . 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 5951 . . . . . . . 8  |-  ( ( F : A --> B  /\  x  e.  A )  ->  ( ( R  o.  F ) `  x
)  =  ( R `
 ( F `  x ) ) )
74, 5, 6syl2anc 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 5951 . . . . . . . 8  |-  ( ( F : A --> B  /\  y  e.  A )  ->  ( ( R  o.  F ) `  y
)  =  ( R `
 ( F `  y ) ) )
104, 8, 9syl2anc 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 ) ) )
113, 7, 103eqtr4d 2518 . . . . . 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 )
)
1312fveq1d 5874 . . . . . 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 5874 . . . . . 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 2516 . . . . 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 6098 . . . . . 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 6098 . . . . . 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 2516 . . . 4  |-  ( ( ( F : A --> B  /\  ( R  o.  F )  =  (  _I  |`  A )
)  /\  ( (
x  e.  A  /\  y  e.  A )  /\  ( F `  x
)  =  ( F `
 y ) ) )  ->  x  =  y )
2120expr 615 . . 3  |-  ( ( ( F : A --> B  /\  ( R  o.  F )  =  (  _I  |`  A )
)  /\  ( x  e.  A  /\  y  e.  A ) )  -> 
( ( F `  x )  =  ( F `  y )  ->  x  =  y ) )
2221ralrimivva 2888 . 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 6165 . 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 664 1  |-  ( ( F : A --> B  /\  ( R  o.  F
)  =  (  _I  |`  A ) )  ->  F : A -1-1-> B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2817    _I cid 4796    |` cres 5007    o. ccom 5009   -->wf 5590   -1-1->wf1 5591   ` cfv 5594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fv 5602
This theorem is referenced by:  fcof1od  6196
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