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Theorem dffo4 5343
Description: Alternate definition of an onto mapping. (Contributed by NM, 20-Mar-2007.)
Assertion
Ref Expression
dffo4  |-  ( F : A -onto-> B  <->  ( F : A --> B  /\  A. y  e.  B  E. x  e.  A  x F y ) )
Distinct variable groups:    x, y, A   
x, B, y    x, F, y

Proof of Theorem dffo4
StepHypRef Expression
1 dffo2 5138 . . 3  |-  ( F : A -onto-> B  <->  ( F : A --> B  /\  ran  F  =  B ) )
2 simpl 106 . . . 4  |-  ( ( F : A --> B  /\  ran  F  =  B )  ->  F : A --> B )
3 vex 2577 . . . . . . . . . 10  |-  y  e. 
_V
43elrn 4605 . . . . . . . . 9  |-  ( y  e.  ran  F  <->  E. x  x F y )
5 eleq2 2117 . . . . . . . . 9  |-  ( ran 
F  =  B  -> 
( y  e.  ran  F  <-> 
y  e.  B ) )
64, 5syl5bbr 187 . . . . . . . 8  |-  ( ran 
F  =  B  -> 
( E. x  x F y  <->  y  e.  B ) )
76biimpar 285 . . . . . . 7  |-  ( ( ran  F  =  B  /\  y  e.  B
)  ->  E. x  x F y )
87adantll 453 . . . . . 6  |-  ( ( ( F : A --> B  /\  ran  F  =  B )  /\  y  e.  B )  ->  E. x  x F y )
9 ffn 5074 . . . . . . . . . . 11  |-  ( F : A --> B  ->  F  Fn  A )
10 fnbr 5029 . . . . . . . . . . . 12  |-  ( ( F  Fn  A  /\  x F y )  ->  x  e.  A )
1110ex 112 . . . . . . . . . . 11  |-  ( F  Fn  A  ->  (
x F y  ->  x  e.  A )
)
129, 11syl 14 . . . . . . . . . 10  |-  ( F : A --> B  -> 
( x F y  ->  x  e.  A
) )
1312ancrd 313 . . . . . . . . 9  |-  ( F : A --> B  -> 
( x F y  ->  ( x  e.  A  /\  x F y ) ) )
1413eximdv 1776 . . . . . . . 8  |-  ( F : A --> B  -> 
( E. x  x F y  ->  E. x
( x  e.  A  /\  x F y ) ) )
15 df-rex 2329 . . . . . . . 8  |-  ( E. x  e.  A  x F y  <->  E. x
( x  e.  A  /\  x F y ) )
1614, 15syl6ibr 155 . . . . . . 7  |-  ( F : A --> B  -> 
( E. x  x F y  ->  E. x  e.  A  x F
y ) )
1716ad2antrr 465 . . . . . 6  |-  ( ( ( F : A --> B  /\  ran  F  =  B )  /\  y  e.  B )  ->  ( E. x  x F
y  ->  E. x  e.  A  x F
y ) )
188, 17mpd 13 . . . . 5  |-  ( ( ( F : A --> B  /\  ran  F  =  B )  /\  y  e.  B )  ->  E. x  e.  A  x F
y )
1918ralrimiva 2409 . . . 4  |-  ( ( F : A --> B  /\  ran  F  =  B )  ->  A. y  e.  B  E. x  e.  A  x F y )
202, 19jca 294 . . 3  |-  ( ( F : A --> B  /\  ran  F  =  B )  ->  ( F : A
--> B  /\  A. y  e.  B  E. x  e.  A  x F
y ) )
211, 20sylbi 118 . 2  |-  ( F : A -onto-> B  -> 
( F : A --> B  /\  A. y  e.  B  E. x  e.  A  x F y ) )
22 fnbrfvb 5242 . . . . . . . . 9  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( ( F `  x )  =  y  <-> 
x F y ) )
2322biimprd 151 . . . . . . . 8  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( x F y  ->  ( F `  x )  =  y ) )
24 eqcom 2058 . . . . . . . 8  |-  ( ( F `  x )  =  y  <->  y  =  ( F `  x ) )
2523, 24syl6ib 154 . . . . . . 7  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( x F y  ->  y  =  ( F `  x ) ) )
269, 25sylan 271 . . . . . 6  |-  ( ( F : A --> B  /\  x  e.  A )  ->  ( x F y  ->  y  =  ( F `  x ) ) )
2726reximdva 2438 . . . . 5  |-  ( F : A --> B  -> 
( E. x  e.  A  x F y  ->  E. x  e.  A  y  =  ( F `  x ) ) )
2827ralimdv 2405 . . . 4  |-  ( F : A --> B  -> 
( A. y  e.  B  E. x  e.  A  x F y  ->  A. y  e.  B  E. x  e.  A  y  =  ( F `  x ) ) )
2928imdistani 427 . . 3  |-  ( ( F : A --> B  /\  A. y  e.  B  E. x  e.  A  x F y )  -> 
( F : A --> B  /\  A. y  e.  B  E. x  e.  A  y  =  ( F `  x ) ) )
30 dffo3 5342 . . 3  |-  ( F : A -onto-> B  <->  ( F : A --> B  /\  A. y  e.  B  E. x  e.  A  y  =  ( F `  x ) ) )
3129, 30sylibr 141 . 2  |-  ( ( F : A --> B  /\  A. y  e.  B  E. x  e.  A  x F y )  ->  F : A -onto-> B )
3221, 31impbii 121 1  |-  ( F : A -onto-> B  <->  ( F : A --> B  /\  A. y  e.  B  E. x  e.  A  x F y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102    = wceq 1259   E.wex 1397    e. wcel 1409   A.wral 2323   E.wrex 2324   class class class wbr 3792   ran crn 4374    Fn wfn 4925   -->wf 4926   -onto->wfo 4928   ` cfv 4930
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2788  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-fo 4936  df-fv 4938
This theorem is referenced by:  dffo5  5344
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