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Theorem fcofo 5718
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 5583 . . . . 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 5446 . . . . 5  |-  ( ( ( F : A --> B  /\  S : B --> A  /\  ( F  o.  S )  =  (  _I  |`  B )
)  /\  y  e.  B )  ->  (
( F  o.  S
) `  y )  =  ( (  _I  |`  B ) `  y
) )
6 fvco3 5516 . . . . . 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 5631 . . . . . 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 5444 . . . . . 6  |-  ( x  =  ( S `  y )  ->  ( F `  x )  =  ( F `  ( S `  y ) ) )
1211eqeq2d 2267 . . . . 5  |-  ( x  =  ( S `  y )  ->  (
y  =  ( F `
 x )  <->  y  =  ( F `  ( S `
 y ) ) ) )
1312rcla4ev 2852 . . . 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 2599 . 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 5595 . 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 4262    |` cres 4649    o. ccom 4651   -->wf 4655   -onto->wfo 4657   ` cfv 4659
This theorem is referenced by:  fcof1o  5723  surjsec2  24473
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 4101  ax-nul 4109  ax-pr 4172  ax-un 4470
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 2521  df-rex 2522  df-rab 2525  df-v 2759  df-sbc 2953  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-br 3984  df-opab 4038  df-mpt 4039  df-id 4267  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-fo 4673  df-fv 4675
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