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Theorem ovelrn 5677
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 5674 . . 3  |-  ( F  Fn  ( A  X.  B )  ->  ran  F  =  { z  |  E. x  e.  A  E. y  e.  B  z  =  ( x F y ) } )
21eleq2d 2123 . 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 2583 . . . 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 5566 . . . . . 6  |-  ( ( F  Fn  ( A  X.  B )  /\  x  e.  A  /\  y  e.  B )  ->  ( x F y )  e.  _V )
6 eleq1 2116 . . . . . 6  |-  ( C  =  ( x F y )  ->  ( C  e.  _V  <->  ( x F y )  e. 
_V ) )
75, 6syl5ibrcom 150 . . . . 5  |-  ( ( F  Fn  ( A  X.  B )  /\  x  e.  A  /\  y  e.  B )  ->  ( C  =  ( x F y )  ->  C  e.  _V ) )
873expb 1116 . . . 4  |-  ( ( F  Fn  ( A  X.  B )  /\  ( x  e.  A  /\  y  e.  B
) )  ->  ( C  =  ( x F y )  ->  C  e.  _V )
)
98rexlimdvva 2457 . . 3  |-  ( F  Fn  ( A  X.  B )  ->  ( E. x  e.  A  E. y  e.  B  C  =  ( x F y )  ->  C  e.  _V )
)
10 eqeq1 2062 . . . . . 6  |-  ( z  =  C  ->  (
z  =  ( x F y )  <->  C  =  ( x F y ) ) )
11102rexbidv 2366 . . . . 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 2711 . . . 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 631 . 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 181 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 102    /\ w3a 896    = wceq 1259    e. wcel 1409   {cab 2042   E.wrex 2324   _Vcvv 2574    X. cxp 4371   ran crn 4374    Fn wfn 4925  (class class class)co 5540
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2788  df-csb 2881  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-iota 4895  df-fun 4932  df-fn 4933  df-fv 4938  df-ov 5543
This theorem is referenced by: (None)
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