MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dffo4 Structured version   Unicode version

Theorem dffo4 5885
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 5657 . . 3  |-  ( F : A -onto-> B  <->  ( F : A --> B  /\  ran  F  =  B ) )
2 simpl 444 . . . 4  |-  ( ( F : A --> B  /\  ran  F  =  B )  ->  F : A --> B )
3 vex 2959 . . . . . . . . . 10  |-  y  e. 
_V
43elrn 5110 . . . . . . . . 9  |-  ( y  e.  ran  F  <->  E. x  x F y )
5 eleq2 2497 . . . . . . . . 9  |-  ( ran 
F  =  B  -> 
( y  e.  ran  F  <-> 
y  e.  B ) )
64, 5syl5bbr 251 . . . . . . . 8  |-  ( ran 
F  =  B  -> 
( E. x  x F y  <->  y  e.  B ) )
76biimpar 472 . . . . . . 7  |-  ( ( ran  F  =  B  /\  y  e.  B
)  ->  E. x  x F y )
87adantll 695 . . . . . 6  |-  ( ( ( F : A --> B  /\  ran  F  =  B )  /\  y  e.  B )  ->  E. x  x F y )
9 ffn 5591 . . . . . . . . . . 11  |-  ( F : A --> B  ->  F  Fn  A )
10 fnbr 5547 . . . . . . . . . . . 12  |-  ( ( F  Fn  A  /\  x F y )  ->  x  e.  A )
1110ex 424 . . . . . . . . . . 11  |-  ( F  Fn  A  ->  (
x F y  ->  x  e.  A )
)
129, 11syl 16 . . . . . . . . . 10  |-  ( F : A --> B  -> 
( x F y  ->  x  e.  A
) )
1312ancrd 538 . . . . . . . . 9  |-  ( F : A --> B  -> 
( x F y  ->  ( x  e.  A  /\  x F y ) ) )
1413eximdv 1632 . . . . . . . 8  |-  ( F : A --> B  -> 
( E. x  x F y  ->  E. x
( x  e.  A  /\  x F y ) ) )
15 df-rex 2711 . . . . . . . 8  |-  ( E. x  e.  A  x F y  <->  E. x
( x  e.  A  /\  x F y ) )
1614, 15syl6ibr 219 . . . . . . 7  |-  ( F : A --> B  -> 
( E. x  x F y  ->  E. x  e.  A  x F
y ) )
1716ad2antrr 707 . . . . . 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 2789 . . . 4  |-  ( ( F : A --> B  /\  ran  F  =  B )  ->  A. y  e.  B  E. x  e.  A  x F y )
202, 19jca 519 . . 3  |-  ( ( F : A --> B  /\  ran  F  =  B )  ->  ( F : A
--> B  /\  A. y  e.  B  E. x  e.  A  x F
y ) )
211, 20sylbi 188 . 2  |-  ( F : A -onto-> B  -> 
( F : A --> B  /\  A. y  e.  B  E. x  e.  A  x F y ) )
22 fnbrfvb 5767 . . . . . . . . 9  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( ( F `  x )  =  y  <-> 
x F y ) )
2322biimprd 215 . . . . . . . 8  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( x F y  ->  ( F `  x )  =  y ) )
24 eqcom 2438 . . . . . . . 8  |-  ( ( F `  x )  =  y  <->  y  =  ( F `  x ) )
2523, 24syl6ib 218 . . . . . . 7  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( x F y  ->  y  =  ( F `  x ) ) )
269, 25sylan 458 . . . . . 6  |-  ( ( F : A --> B  /\  x  e.  A )  ->  ( x F y  ->  y  =  ( F `  x ) ) )
2726reximdva 2818 . . . . 5  |-  ( F : A --> B  -> 
( E. x  e.  A  x F y  ->  E. x  e.  A  y  =  ( F `  x ) ) )
2827ralimdv 2785 . . . 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 672 . . 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 5884 . . 3  |-  ( F : A -onto-> B  <->  ( F : A --> B  /\  A. y  e.  B  E. x  e.  A  y  =  ( F `  x ) ) )
3129, 30sylibr 204 . 2  |-  ( ( F : A --> B  /\  A. y  e.  B  E. x  e.  A  x F y )  ->  F : A -onto-> B )
3221, 31impbii 181 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 177    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725   A.wral 2705   E.wrex 2706   class class class wbr 4212   ran crn 4879    Fn wfn 5449   -->wf 5450   -onto->wfo 5452   ` cfv 5454
This theorem is referenced by:  dffo5  5886  exfo  5887  brdom3  8406
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fo 5460  df-fv 5462
  Copyright terms: Public domain W3C validator