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Theorem dffo4 5676
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 5455 . . 3  |-  ( F : A -onto-> B  <->  ( F : A --> B  /\  ran  F  =  B ) )
2 simpl 443 . . . 4  |-  ( ( F : A --> B  /\  ran  F  =  B )  ->  F : A --> B )
3 vex 2791 . . . . . . . . . 10  |-  y  e. 
_V
43elrn 4919 . . . . . . . . 9  |-  ( y  e.  ran  F  <->  E. x  x F y )
5 eleq2 2344 . . . . . . . . 9  |-  ( ran 
F  =  B  -> 
( y  e.  ran  F  <-> 
y  e.  B ) )
64, 5syl5bbr 250 . . . . . . . 8  |-  ( ran 
F  =  B  -> 
( E. x  x F y  <->  y  e.  B ) )
76biimpar 471 . . . . . . 7  |-  ( ( ran  F  =  B  /\  y  e.  B
)  ->  E. x  x F y )
87adantll 694 . . . . . 6  |-  ( ( ( F : A --> B  /\  ran  F  =  B )  /\  y  e.  B )  ->  E. x  x F y )
9 ffn 5389 . . . . . . . . . . 11  |-  ( F : A --> B  ->  F  Fn  A )
10 fnbr 5346 . . . . . . . . . . . 12  |-  ( ( F  Fn  A  /\  x F y )  ->  x  e.  A )
1110ex 423 . . . . . . . . . . 11  |-  ( F  Fn  A  ->  (
x F y  ->  x  e.  A )
)
129, 11syl 15 . . . . . . . . . 10  |-  ( F : A --> B  -> 
( x F y  ->  x  e.  A
) )
1312ancrd 537 . . . . . . . . 9  |-  ( F : A --> B  -> 
( x F y  ->  ( x  e.  A  /\  x F y ) ) )
1413eximdv 1608 . . . . . . . 8  |-  ( F : A --> B  -> 
( E. x  x F y  ->  E. x
( x  e.  A  /\  x F y ) ) )
15 df-rex 2549 . . . . . . . 8  |-  ( E. x  e.  A  x F y  <->  E. x
( x  e.  A  /\  x F y ) )
1614, 15syl6ibr 218 . . . . . . 7  |-  ( F : A --> B  -> 
( E. x  x F y  ->  E. x  e.  A  x F
y ) )
1716ad2antrr 706 . . . . . 6  |-  ( ( ( F : A --> B  /\  ran  F  =  B )  /\  y  e.  B )  ->  ( E. x  x F
y  ->  E. x  e.  A  x F
y ) )
188, 17mpd 14 . . . . 5  |-  ( ( ( F : A --> B  /\  ran  F  =  B )  /\  y  e.  B )  ->  E. x  e.  A  x F
y )
1918ralrimiva 2626 . . . 4  |-  ( ( F : A --> B  /\  ran  F  =  B )  ->  A. y  e.  B  E. x  e.  A  x F y )
202, 19jca 518 . . 3  |-  ( ( F : A --> B  /\  ran  F  =  B )  ->  ( F : A
--> B  /\  A. y  e.  B  E. x  e.  A  x F
y ) )
211, 20sylbi 187 . 2  |-  ( F : A -onto-> B  -> 
( F : A --> B  /\  A. y  e.  B  E. x  e.  A  x F y ) )
22 fnbrfvb 5563 . . . . . . . . 9  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( ( F `  x )  =  y  <-> 
x F y ) )
2322biimprd 214 . . . . . . . 8  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( x F y  ->  ( F `  x )  =  y ) )
24 eqcom 2285 . . . . . . . 8  |-  ( ( F `  x )  =  y  <->  y  =  ( F `  x ) )
2523, 24syl6ib 217 . . . . . . 7  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( x F y  ->  y  =  ( F `  x ) ) )
269, 25sylan 457 . . . . . 6  |-  ( ( F : A --> B  /\  x  e.  A )  ->  ( x F y  ->  y  =  ( F `  x ) ) )
2726reximdva 2655 . . . . 5  |-  ( F : A --> B  -> 
( E. x  e.  A  x F y  ->  E. x  e.  A  y  =  ( F `  x ) ) )
2827ralimdv 2622 . . . 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 671 . . 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 5675 . . 3  |-  ( F : A -onto-> B  <->  ( F : A --> B  /\  A. y  e.  B  E. x  e.  A  y  =  ( F `  x ) ) )
3129, 30sylibr 203 . 2  |-  ( ( F : A --> B  /\  A. y  e.  B  E. x  e.  A  x F y )  ->  F : A -onto-> B )
3221, 31impbii 180 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    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   class class class wbr 4023   ran crn 4690    Fn wfn 5250   -->wf 5251   -onto->wfo 5253   ` cfv 5255
This theorem is referenced by:  dffo5  5677  exfo  5678  brdom3  8153
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263
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