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Theorem dffo4 5536
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 5319 . . 3  |-  ( F : A -onto-> B  <->  ( F : A --> B  /\  ran  F  =  B ) )
2 simpl 108 . . . 4  |-  ( ( F : A --> B  /\  ran  F  =  B )  ->  F : A --> B )
3 vex 2663 . . . . . . . . . 10  |-  y  e. 
_V
43elrn 4752 . . . . . . . . 9  |-  ( y  e.  ran  F  <->  E. x  x F y )
5 eleq2 2181 . . . . . . . . 9  |-  ( ran 
F  =  B  -> 
( y  e.  ran  F  <-> 
y  e.  B ) )
64, 5syl5bbr 193 . . . . . . . 8  |-  ( ran 
F  =  B  -> 
( E. x  x F y  <->  y  e.  B ) )
76biimpar 295 . . . . . . 7  |-  ( ( ran  F  =  B  /\  y  e.  B
)  ->  E. x  x F y )
87adantll 467 . . . . . 6  |-  ( ( ( F : A --> B  /\  ran  F  =  B )  /\  y  e.  B )  ->  E. x  x F y )
9 ffn 5242 . . . . . . . . . . 11  |-  ( F : A --> B  ->  F  Fn  A )
10 fnbr 5195 . . . . . . . . . . . 12  |-  ( ( F  Fn  A  /\  x F y )  ->  x  e.  A )
1110ex 114 . . . . . . . . . . 11  |-  ( F  Fn  A  ->  (
x F y  ->  x  e.  A )
)
129, 11syl 14 . . . . . . . . . 10  |-  ( F : A --> B  -> 
( x F y  ->  x  e.  A
) )
1312ancrd 324 . . . . . . . . 9  |-  ( F : A --> B  -> 
( x F y  ->  ( x  e.  A  /\  x F y ) ) )
1413eximdv 1836 . . . . . . . 8  |-  ( F : A --> B  -> 
( E. x  x F y  ->  E. x
( x  e.  A  /\  x F y ) ) )
15 df-rex 2399 . . . . . . . 8  |-  ( E. x  e.  A  x F y  <->  E. x
( x  e.  A  /\  x F y ) )
1614, 15syl6ibr 161 . . . . . . 7  |-  ( F : A --> B  -> 
( E. x  x F y  ->  E. x  e.  A  x F
y ) )
1716ad2antrr 479 . . . . . 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 2482 . . . 4  |-  ( ( F : A --> B  /\  ran  F  =  B )  ->  A. y  e.  B  E. x  e.  A  x F y )
202, 19jca 304 . . 3  |-  ( ( F : A --> B  /\  ran  F  =  B )  ->  ( F : A
--> B  /\  A. y  e.  B  E. x  e.  A  x F
y ) )
211, 20sylbi 120 . 2  |-  ( F : A -onto-> B  -> 
( F : A --> B  /\  A. y  e.  B  E. x  e.  A  x F y ) )
22 fnbrfvb 5430 . . . . . . . . 9  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( ( F `  x )  =  y  <-> 
x F y ) )
2322biimprd 157 . . . . . . . 8  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( x F y  ->  ( F `  x )  =  y ) )
24 eqcom 2119 . . . . . . . 8  |-  ( ( F `  x )  =  y  <->  y  =  ( F `  x ) )
2523, 24syl6ib 160 . . . . . . 7  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( x F y  ->  y  =  ( F `  x ) ) )
269, 25sylan 281 . . . . . 6  |-  ( ( F : A --> B  /\  x  e.  A )  ->  ( x F y  ->  y  =  ( F `  x ) ) )
2726reximdva 2511 . . . . 5  |-  ( F : A --> B  -> 
( E. x  e.  A  x F y  ->  E. x  e.  A  y  =  ( F `  x ) ) )
2827ralimdv 2477 . . . 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 441 . . 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 5535 . . 3  |-  ( F : A -onto-> B  <->  ( F : A --> B  /\  A. y  e.  B  E. x  e.  A  y  =  ( F `  x ) ) )
3129, 30sylibr 133 . 2  |-  ( ( F : A --> B  /\  A. y  e.  B  E. x  e.  A  x F y )  ->  F : A -onto-> B )
3221, 31impbii 125 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 103    <-> wb 104    = wceq 1316   E.wex 1453    e. wcel 1465   A.wral 2393   E.wrex 2394   class class class wbr 3899   ran crn 4510    Fn wfn 5088   -->wf 5089   -onto->wfo 5091   ` cfv 5093
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-fo 5099  df-fv 5101
This theorem is referenced by:  dffo5  5537
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