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Theorem fcofo 6172
Description: An application is surjective if a section exists. Proposition 8 of [BourbakiEns] p. E.II.18. (Contributed by FL, 17-Nov-2011.) (Proof shortened by Mario Carneiro, 27-Dec-2014.)
Assertion
Ref Expression
fcofo  |-  ( ( F : A --> B  /\  S : B --> A  /\  ( F  o.  S
)  =  (  _I  |`  B ) )  ->  F : A -onto-> B )

Proof of Theorem fcofo
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 991 . 2  |-  ( ( F : A --> B  /\  S : B --> A  /\  ( F  o.  S
)  =  (  _I  |`  B ) )  ->  F : A --> B )
2 ffvelrn 6012 . . . . 5  |-  ( ( S : B --> A  /\  y  e.  B )  ->  ( S `  y
)  e.  A )
323ad2antl2 1154 . . . 4  |-  ( ( ( F : A --> B  /\  S : B --> A  /\  ( F  o.  S )  =  (  _I  |`  B )
)  /\  y  e.  B )  ->  ( S `  y )  e.  A )
4 simpl3 996 . . . . . 6  |-  ( ( ( F : A --> B  /\  S : B --> A  /\  ( F  o.  S )  =  (  _I  |`  B )
)  /\  y  e.  B )  ->  ( F  o.  S )  =  (  _I  |`  B ) )
54fveq1d 5861 . . . . 5  |-  ( ( ( F : A --> B  /\  S : B --> A  /\  ( F  o.  S )  =  (  _I  |`  B )
)  /\  y  e.  B )  ->  (
( F  o.  S
) `  y )  =  ( (  _I  |`  B ) `  y
) )
6 fvco3 5937 . . . . . 6  |-  ( ( S : B --> A  /\  y  e.  B )  ->  ( ( F  o.  S ) `  y
)  =  ( F `
 ( S `  y ) ) )
763ad2antl2 1154 . . . . 5  |-  ( ( ( F : A --> B  /\  S : B --> A  /\  ( F  o.  S )  =  (  _I  |`  B )
)  /\  y  e.  B )  ->  (
( F  o.  S
) `  y )  =  ( F `  ( S `  y ) ) )
8 fvresi 6080 . . . . . 6  |-  ( y  e.  B  ->  (
(  _I  |`  B ) `
 y )  =  y )
98adantl 466 . . . . 5  |-  ( ( ( F : A --> B  /\  S : B --> A  /\  ( F  o.  S )  =  (  _I  |`  B )
)  /\  y  e.  B )  ->  (
(  _I  |`  B ) `
 y )  =  y )
105, 7, 93eqtr3rd 2512 . . . 4  |-  ( ( ( F : A --> B  /\  S : B --> A  /\  ( F  o.  S )  =  (  _I  |`  B )
)  /\  y  e.  B )  ->  y  =  ( F `  ( S `  y ) ) )
11 fveq2 5859 . . . . . 6  |-  ( x  =  ( S `  y )  ->  ( F `  x )  =  ( F `  ( S `  y ) ) )
1211eqeq2d 2476 . . . . 5  |-  ( x  =  ( S `  y )  ->  (
y  =  ( F `
 x )  <->  y  =  ( F `  ( S `
 y ) ) ) )
1312rspcev 3209 . . . 4  |-  ( ( ( S `  y
)  e.  A  /\  y  =  ( F `  ( S `  y
) ) )  ->  E. x  e.  A  y  =  ( F `  x ) )
143, 10, 13syl2anc 661 . . 3  |-  ( ( ( F : A --> B  /\  S : B --> A  /\  ( F  o.  S )  =  (  _I  |`  B )
)  /\  y  e.  B )  ->  E. x  e.  A  y  =  ( F `  x ) )
1514ralrimiva 2873 . 2  |-  ( ( F : A --> B  /\  S : B --> A  /\  ( F  o.  S
)  =  (  _I  |`  B ) )  ->  A. y  e.  B  E. x  e.  A  y  =  ( F `  x ) )
16 dffo3 6029 . 2  |-  ( F : A -onto-> B  <->  ( F : A --> B  /\  A. y  e.  B  E. x  e.  A  y  =  ( F `  x ) ) )
171, 15, 16sylanbrc 664 1  |-  ( ( F : A --> B  /\  S : B --> A  /\  ( F  o.  S
)  =  (  _I  |`  B ) )  ->  F : A -onto-> B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   A.wral 2809   E.wrex 2810    _I cid 4785    |` cres 4996    o. ccom 4998   -->wf 5577   -onto->wfo 5579   ` cfv 5581
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-fo 5587  df-fv 5589
This theorem is referenced by:  fcof1od  6178
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