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Theorem dffo4 6032
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 5789 . . 3  |-  ( F : A -onto-> B  <->  ( F : A --> B  /\  ran  F  =  B ) )
2 simpl 457 . . . 4  |-  ( ( F : A --> B  /\  ran  F  =  B )  ->  F : A --> B )
3 vex 3098 . . . . . . . . . 10  |-  y  e. 
_V
43elrn 5233 . . . . . . . . 9  |-  ( y  e.  ran  F  <->  E. x  x F y )
5 eleq2 2516 . . . . . . . . 9  |-  ( ran 
F  =  B  -> 
( y  e.  ran  F  <-> 
y  e.  B ) )
64, 5syl5bbr 259 . . . . . . . 8  |-  ( ran 
F  =  B  -> 
( E. x  x F y  <->  y  e.  B ) )
76biimpar 485 . . . . . . 7  |-  ( ( ran  F  =  B  /\  y  e.  B
)  ->  E. x  x F y )
87adantll 713 . . . . . 6  |-  ( ( ( F : A --> B  /\  ran  F  =  B )  /\  y  e.  B )  ->  E. x  x F y )
9 ffn 5721 . . . . . . . . . . 11  |-  ( F : A --> B  ->  F  Fn  A )
10 fnbr 5673 . . . . . . . . . . . 12  |-  ( ( F  Fn  A  /\  x F y )  ->  x  e.  A )
1110ex 434 . . . . . . . . . . 11  |-  ( F  Fn  A  ->  (
x F y  ->  x  e.  A )
)
129, 11syl 16 . . . . . . . . . 10  |-  ( F : A --> B  -> 
( x F y  ->  x  e.  A
) )
1312ancrd 554 . . . . . . . . 9  |-  ( F : A --> B  -> 
( x F y  ->  ( x  e.  A  /\  x F y ) ) )
1413eximdv 1697 . . . . . . . 8  |-  ( F : A --> B  -> 
( E. x  x F y  ->  E. x
( x  e.  A  /\  x F y ) ) )
15 df-rex 2799 . . . . . . . 8  |-  ( E. x  e.  A  x F y  <->  E. x
( x  e.  A  /\  x F y ) )
1614, 15syl6ibr 227 . . . . . . 7  |-  ( F : A --> B  -> 
( E. x  x F y  ->  E. x  e.  A  x F
y ) )
1716ad2antrr 725 . . . . . 6  |-  ( ( ( F : A --> B  /\  ran  F  =  B )  /\  y  e.  B )  ->  ( E. x  x F
y  ->  E. x  e.  A  x F
y ) )
188, 17mpd 15 . . . . 5  |-  ( ( ( F : A --> B  /\  ran  F  =  B )  /\  y  e.  B )  ->  E. x  e.  A  x F
y )
1918ralrimiva 2857 . . . 4  |-  ( ( F : A --> B  /\  ran  F  =  B )  ->  A. y  e.  B  E. x  e.  A  x F y )
202, 19jca 532 . . 3  |-  ( ( F : A --> B  /\  ran  F  =  B )  ->  ( F : A
--> B  /\  A. y  e.  B  E. x  e.  A  x F
y ) )
211, 20sylbi 195 . 2  |-  ( F : A -onto-> B  -> 
( F : A --> B  /\  A. y  e.  B  E. x  e.  A  x F y ) )
22 fnbrfvb 5898 . . . . . . . . 9  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( ( F `  x )  =  y  <-> 
x F y ) )
2322biimprd 223 . . . . . . . 8  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( x F y  ->  ( F `  x )  =  y ) )
24 eqcom 2452 . . . . . . . 8  |-  ( ( F `  x )  =  y  <->  y  =  ( F `  x ) )
2523, 24syl6ib 226 . . . . . . 7  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( x F y  ->  y  =  ( F `  x ) ) )
269, 25sylan 471 . . . . . 6  |-  ( ( F : A --> B  /\  x  e.  A )  ->  ( x F y  ->  y  =  ( F `  x ) ) )
2726reximdva 2918 . . . . 5  |-  ( F : A --> B  -> 
( E. x  e.  A  x F y  ->  E. x  e.  A  y  =  ( F `  x ) ) )
2827ralimdv 2853 . . . 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 690 . . 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 6031 . . 3  |-  ( F : A -onto-> B  <->  ( F : A --> B  /\  A. y  e.  B  E. x  e.  A  y  =  ( F `  x ) ) )
3129, 30sylibr 212 . 2  |-  ( ( F : A --> B  /\  A. y  e.  B  E. x  e.  A  x F y )  ->  F : A -onto-> B )
3221, 31impbii 188 1  |-  ( F : A -onto-> B  <->  ( F : A --> B  /\  A. y  e.  B  E. x  e.  A  x F y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1383   E.wex 1599    e. wcel 1804   A.wral 2793   E.wrex 2794   class class class wbr 4437   ran crn 4990    Fn wfn 5573   -->wf 5574   -onto->wfo 5576   ` cfv 5578
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-fo 5584  df-fv 5586
This theorem is referenced by:  dffo5  6033  exfo  6034  brdom3  8909
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