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Theorem ovelrn 5793
Description: A member of an operation's range is a value of the operation. (Contributed by NM, 7-Feb-2007.) (Revised by Mario Carneiro, 30-Jan-2014.)
Assertion
Ref Expression
ovelrn  |-  ( F  Fn  ( A  X.  B )  ->  ( C  e.  ran  F  <->  E. x  e.  A  E. y  e.  B  C  =  ( x F y ) ) )
Distinct variable groups:    x, y, A   
x, B, y    x, C, y    x, F, y

Proof of Theorem ovelrn
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 fnrnov 5790 . . 3  |-  ( F  Fn  ( A  X.  B )  ->  ran  F  =  { z  |  E. x  e.  A  E. y  e.  B  z  =  ( x F y ) } )
21eleq2d 2157 . 2  |-  ( F  Fn  ( A  X.  B )  ->  ( C  e.  ran  F  <->  C  e.  { z  |  E. x  e.  A  E. y  e.  B  z  =  ( x F y ) } ) )
3 elex 2630 . . . 4  |-  ( C  e.  { z  |  E. x  e.  A  E. y  e.  B  z  =  ( x F y ) }  ->  C  e.  _V )
43a1i 9 . . 3  |-  ( F  Fn  ( A  X.  B )  ->  ( C  e.  { z  |  E. x  e.  A  E. y  e.  B  z  =  ( x F y ) }  ->  C  e.  _V ) )
5 fnovex 5682 . . . . . 6  |-  ( ( F  Fn  ( A  X.  B )  /\  x  e.  A  /\  y  e.  B )  ->  ( x F y )  e.  _V )
6 eleq1 2150 . . . . . 6  |-  ( C  =  ( x F y )  ->  ( C  e.  _V  <->  ( x F y )  e. 
_V ) )
75, 6syl5ibrcom 155 . . . . 5  |-  ( ( F  Fn  ( A  X.  B )  /\  x  e.  A  /\  y  e.  B )  ->  ( C  =  ( x F y )  ->  C  e.  _V ) )
873expb 1144 . . . 4  |-  ( ( F  Fn  ( A  X.  B )  /\  ( x  e.  A  /\  y  e.  B
) )  ->  ( C  =  ( x F y )  ->  C  e.  _V )
)
98rexlimdvva 2496 . . 3  |-  ( F  Fn  ( A  X.  B )  ->  ( E. x  e.  A  E. y  e.  B  C  =  ( x F y )  ->  C  e.  _V )
)
10 eqeq1 2094 . . . . . 6  |-  ( z  =  C  ->  (
z  =  ( x F y )  <->  C  =  ( x F y ) ) )
11102rexbidv 2403 . . . . 5  |-  ( z  =  C  ->  ( E. x  e.  A  E. y  e.  B  z  =  ( x F y )  <->  E. x  e.  A  E. y  e.  B  C  =  ( x F y ) ) )
1211elabg 2761 . . . 4  |-  ( C  e.  _V  ->  ( C  e.  { z  |  E. x  e.  A  E. y  e.  B  z  =  ( x F y ) }  <->  E. x  e.  A  E. y  e.  B  C  =  ( x F y ) ) )
1312a1i 9 . . 3  |-  ( F  Fn  ( A  X.  B )  ->  ( C  e.  _V  ->  ( C  e.  { z  |  E. x  e.  A  E. y  e.  B  z  =  ( x F y ) }  <->  E. x  e.  A  E. y  e.  B  C  =  ( x F y ) ) ) )
144, 9, 13pm5.21ndd 656 . 2  |-  ( F  Fn  ( A  X.  B )  ->  ( C  e.  { z  |  E. x  e.  A  E. y  e.  B  z  =  ( x F y ) }  <->  E. x  e.  A  E. y  e.  B  C  =  ( x F y ) ) )
152, 14bitrd 186 1  |-  ( F  Fn  ( A  X.  B )  ->  ( C  e.  ran  F  <->  E. x  e.  A  E. y  e.  B  C  =  ( x F y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    /\ w3a 924    = wceq 1289    e. wcel 1438   {cab 2074   E.wrex 2360   _Vcvv 2619    X. cxp 4436   ran crn 4439    Fn wfn 5010  (class class class)co 5652
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-csb 2934  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-iota 4980  df-fun 5017  df-fn 5018  df-fv 5023  df-ov 5655
This theorem is referenced by: (None)
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