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Theorem fcofo 5476
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 939 . 2  |-  ( ( F : A --> B  /\  S : B --> A  /\  ( F  o.  S
)  =  (  _I  |`  B ) )  ->  F : A --> B )
2 ffvelrn 5353 . . . . 5  |-  ( ( S : B --> A  /\  y  e.  B )  ->  ( S `  y
)  e.  A )
323ad2antl2 1102 . . . 4  |-  ( ( ( F : A --> B  /\  S : B --> A  /\  ( F  o.  S )  =  (  _I  |`  B )
)  /\  y  e.  B )  ->  ( S `  y )  e.  A )
4 simpl3 944 . . . . . 6  |-  ( ( ( F : A --> B  /\  S : B --> A  /\  ( F  o.  S )  =  (  _I  |`  B )
)  /\  y  e.  B )  ->  ( F  o.  S )  =  (  _I  |`  B ) )
54fveq1d 5232 . . . . 5  |-  ( ( ( F : A --> B  /\  S : B --> A  /\  ( F  o.  S )  =  (  _I  |`  B )
)  /\  y  e.  B )  ->  (
( F  o.  S
) `  y )  =  ( (  _I  |`  B ) `  y
) )
6 fvco3 5297 . . . . . 6  |-  ( ( S : B --> A  /\  y  e.  B )  ->  ( ( F  o.  S ) `  y
)  =  ( F `
 ( S `  y ) ) )
763ad2antl2 1102 . . . . 5  |-  ( ( ( F : A --> B  /\  S : B --> A  /\  ( F  o.  S )  =  (  _I  |`  B )
)  /\  y  e.  B )  ->  (
( F  o.  S
) `  y )  =  ( F `  ( S `  y ) ) )
8 fvresi 5409 . . . . . 6  |-  ( y  e.  B  ->  (
(  _I  |`  B ) `
 y )  =  y )
98adantl 271 . . . . 5  |-  ( ( ( F : A --> B  /\  S : B --> A  /\  ( F  o.  S )  =  (  _I  |`  B )
)  /\  y  e.  B )  ->  (
(  _I  |`  B ) `
 y )  =  y )
105, 7, 93eqtr3rd 2124 . . . 4  |-  ( ( ( F : A --> B  /\  S : B --> A  /\  ( F  o.  S )  =  (  _I  |`  B )
)  /\  y  e.  B )  ->  y  =  ( F `  ( S `  y ) ) )
11 fveq2 5230 . . . . . 6  |-  ( x  =  ( S `  y )  ->  ( F `  x )  =  ( F `  ( S `  y ) ) )
1211eqeq2d 2094 . . . . 5  |-  ( x  =  ( S `  y )  ->  (
y  =  ( F `
 x )  <->  y  =  ( F `  ( S `
 y ) ) ) )
1312rspcev 2710 . . . 4  |-  ( ( ( S `  y
)  e.  A  /\  y  =  ( F `  ( S `  y
) ) )  ->  E. x  e.  A  y  =  ( F `  x ) )
143, 10, 13syl2anc 403 . . 3  |-  ( ( ( F : A --> B  /\  S : B --> A  /\  ( F  o.  S )  =  (  _I  |`  B )
)  /\  y  e.  B )  ->  E. x  e.  A  y  =  ( F `  x ) )
1514ralrimiva 2439 . 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 5367 . 2  |-  ( F : A -onto-> B  <->  ( F : A --> B  /\  A. y  e.  B  E. x  e.  A  y  =  ( F `  x ) ) )
171, 15, 16sylanbrc 408 1  |-  ( ( F : A --> B  /\  S : B --> A  /\  ( F  o.  S
)  =  (  _I  |`  B ) )  ->  F : A -onto-> B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    /\ w3a 920    = wceq 1285    e. wcel 1434   A.wral 2353   E.wrex 2354    _I cid 4071    |` cres 4393    o. ccom 4395   -->wf 4948   -onto->wfo 4950   ` cfv 4952
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-sbc 2825  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-opab 3860  df-mpt 3861  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-fo 4958  df-fv 4960
This theorem is referenced by:  fcof1o  5481
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