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Theorem fcofo 5685
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 981 . 2  |-  ( ( F : A --> B  /\  S : B --> A  /\  ( F  o.  S
)  =  (  _I  |`  B ) )  ->  F : A --> B )
2 ffvelrn 5553 . . . . 5  |-  ( ( S : B --> A  /\  y  e.  B )  ->  ( S `  y
)  e.  A )
323ad2antl2 1144 . . . 4  |-  ( ( ( F : A --> B  /\  S : B --> A  /\  ( F  o.  S )  =  (  _I  |`  B )
)  /\  y  e.  B )  ->  ( S `  y )  e.  A )
4 simpl3 986 . . . . . 6  |-  ( ( ( F : A --> B  /\  S : B --> A  /\  ( F  o.  S )  =  (  _I  |`  B )
)  /\  y  e.  B )  ->  ( F  o.  S )  =  (  _I  |`  B ) )
54fveq1d 5423 . . . . 5  |-  ( ( ( F : A --> B  /\  S : B --> A  /\  ( F  o.  S )  =  (  _I  |`  B )
)  /\  y  e.  B )  ->  (
( F  o.  S
) `  y )  =  ( (  _I  |`  B ) `  y
) )
6 fvco3 5492 . . . . . 6  |-  ( ( S : B --> A  /\  y  e.  B )  ->  ( ( F  o.  S ) `  y
)  =  ( F `
 ( S `  y ) ) )
763ad2antl2 1144 . . . . 5  |-  ( ( ( F : A --> B  /\  S : B --> A  /\  ( F  o.  S )  =  (  _I  |`  B )
)  /\  y  e.  B )  ->  (
( F  o.  S
) `  y )  =  ( F `  ( S `  y ) ) )
8 fvresi 5613 . . . . . 6  |-  ( y  e.  B  ->  (
(  _I  |`  B ) `
 y )  =  y )
98adantl 275 . . . . 5  |-  ( ( ( F : A --> B  /\  S : B --> A  /\  ( F  o.  S )  =  (  _I  |`  B )
)  /\  y  e.  B )  ->  (
(  _I  |`  B ) `
 y )  =  y )
105, 7, 93eqtr3rd 2181 . . . 4  |-  ( ( ( F : A --> B  /\  S : B --> A  /\  ( F  o.  S )  =  (  _I  |`  B )
)  /\  y  e.  B )  ->  y  =  ( F `  ( S `  y ) ) )
11 fveq2 5421 . . . . . 6  |-  ( x  =  ( S `  y )  ->  ( F `  x )  =  ( F `  ( S `  y ) ) )
1211eqeq2d 2151 . . . . 5  |-  ( x  =  ( S `  y )  ->  (
y  =  ( F `
 x )  <->  y  =  ( F `  ( S `
 y ) ) ) )
1312rspcev 2789 . . . 4  |-  ( ( ( S `  y
)  e.  A  /\  y  =  ( F `  ( S `  y
) ) )  ->  E. x  e.  A  y  =  ( F `  x ) )
143, 10, 13syl2anc 408 . . 3  |-  ( ( ( F : A --> B  /\  S : B --> A  /\  ( F  o.  S )  =  (  _I  |`  B )
)  /\  y  e.  B )  ->  E. x  e.  A  y  =  ( F `  x ) )
1514ralrimiva 2505 . 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 5567 . 2  |-  ( F : A -onto-> B  <->  ( F : A --> B  /\  A. y  e.  B  E. x  e.  A  y  =  ( F `  x ) ) )
171, 15, 16sylanbrc 413 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 103    /\ w3a 962    = wceq 1331    e. wcel 1480   A.wral 2416   E.wrex 2417    _I cid 4210    |` cres 4541    o. ccom 4543   -->wf 5119   -onto->wfo 5121   ` cfv 5123
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fo 5129  df-fv 5131
This theorem is referenced by:  fcof1o  5690
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