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Theorem fcofo 5697
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
StepHypRef Expression
1 simp1 960 . 2  |-  ( ( F : A --> B  /\  S : B --> A  /\  ( F  o.  S
)  =  (  _I  |`  B ) )  ->  F : A --> B )
2 ffvelrn 5562 . . . . 5  |-  ( ( S : B --> A  /\  y  e.  B )  ->  ( S `  y
)  e.  A )
323ad2antl2 1123 . . . 4  |-  ( ( ( F : A --> B  /\  S : B --> A  /\  ( F  o.  S )  =  (  _I  |`  B )
)  /\  y  e.  B )  ->  ( S `  y )  e.  A )
4 simpl3 965 . . . . . 6  |-  ( ( ( F : A --> B  /\  S : B --> A  /\  ( F  o.  S )  =  (  _I  |`  B )
)  /\  y  e.  B )  ->  ( F  o.  S )  =  (  _I  |`  B ) )
54fveq1d 5425 . . . . 5  |-  ( ( ( F : A --> B  /\  S : B --> A  /\  ( F  o.  S )  =  (  _I  |`  B )
)  /\  y  e.  B )  ->  (
( F  o.  S
) `  y )  =  ( (  _I  |`  B ) `  y
) )
6 fvco3 5495 . . . . . 6  |-  ( ( S : B --> A  /\  y  e.  B )  ->  ( ( F  o.  S ) `  y
)  =  ( F `
 ( S `  y ) ) )
763ad2antl2 1123 . . . . 5  |-  ( ( ( F : A --> B  /\  S : B --> A  /\  ( F  o.  S )  =  (  _I  |`  B )
)  /\  y  e.  B )  ->  (
( F  o.  S
) `  y )  =  ( F `  ( S `  y ) ) )
8 fvresi 5610 . . . . . 6  |-  ( y  e.  B  ->  (
(  _I  |`  B ) `
 y )  =  y )
98adantl 454 . . . . 5  |-  ( ( ( F : A --> B  /\  S : B --> A  /\  ( F  o.  S )  =  (  _I  |`  B )
)  /\  y  e.  B )  ->  (
(  _I  |`  B ) `
 y )  =  y )
105, 7, 93eqtr3rd 2297 . . . 4  |-  ( ( ( F : A --> B  /\  S : B --> A  /\  ( F  o.  S )  =  (  _I  |`  B )
)  /\  y  e.  B )  ->  y  =  ( F `  ( S `  y ) ) )
11 fveq2 5423 . . . . . 6  |-  ( x  =  ( S `  y )  ->  ( F `  x )  =  ( F `  ( S `  y ) ) )
1211eqeq2d 2267 . . . . 5  |-  ( x  =  ( S `  y )  ->  (
y  =  ( F `
 x )  <->  y  =  ( F `  ( S `
 y ) ) ) )
1312rcla4ev 2835 . . . 4  |-  ( ( ( S `  y
)  e.  A  /\  y  =  ( F `  ( S `  y
) ) )  ->  E. x  e.  A  y  =  ( F `  x ) )
143, 10, 13syl2anc 645 . . 3  |-  ( ( ( F : A --> B  /\  S : B --> A  /\  ( F  o.  S )  =  (  _I  |`  B )
)  /\  y  e.  B )  ->  E. x  e.  A  y  =  ( F `  x ) )
1514ralrimiva 2597 . 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 5574 . 2  |-  ( F : A -onto-> B  <->  ( F : A --> B  /\  A. y  e.  B  E. x  e.  A  y  =  ( F `  x ) ) )
171, 15, 16sylanbrc 648 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 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621   A.wral 2516   E.wrex 2517    _I cid 4241    |` cres 4628    o. ccom 4630   -->wf 4634   -onto->wfo 4636   ` cfv 4638
This theorem is referenced by:  fcof1o  5702  surjsec2  24452
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pr 4152  ax-un 4449
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-rab 2523  df-v 2742  df-sbc 2936  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-br 3964  df-opab 4018  df-mpt 4019  df-id 4246  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-fo 4652  df-fv 4654
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