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Theorem ovelrn 5887
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 5884 . . 3  |-  ( F  Fn  ( A  X.  B )  ->  ran  F  =  { z  |  E. x  e.  A  E. y  e.  B  z  =  ( x F y ) } )
21eleq2d 2187 . 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 2671 . . . 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 5772 . . . . . 6  |-  ( ( F  Fn  ( A  X.  B )  /\  x  e.  A  /\  y  e.  B )  ->  ( x F y )  e.  _V )
6 eleq1 2180 . . . . . 6  |-  ( C  =  ( x F y )  ->  ( C  e.  _V  <->  ( x F y )  e. 
_V ) )
75, 6syl5ibrcom 156 . . . . 5  |-  ( ( F  Fn  ( A  X.  B )  /\  x  e.  A  /\  y  e.  B )  ->  ( C  =  ( x F y )  ->  C  e.  _V ) )
873expb 1167 . . . 4  |-  ( ( F  Fn  ( A  X.  B )  /\  ( x  e.  A  /\  y  e.  B
) )  ->  ( C  =  ( x F y )  ->  C  e.  _V )
)
98rexlimdvva 2534 . . 3  |-  ( F  Fn  ( A  X.  B )  ->  ( E. x  e.  A  E. y  e.  B  C  =  ( x F y )  ->  C  e.  _V )
)
10 eqeq1 2124 . . . . . 6  |-  ( z  =  C  ->  (
z  =  ( x F y )  <->  C  =  ( x F y ) ) )
11102rexbidv 2437 . . . . 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 2803 . . . 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 679 . 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 187 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 104    /\ w3a 947    = wceq 1316    e. wcel 1465   {cab 2103   E.wrex 2394   _Vcvv 2660    X. cxp 4507   ran crn 4510    Fn wfn 5088  (class class class)co 5742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-csb 2976  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-iota 5058  df-fun 5095  df-fn 5096  df-fv 5101  df-ov 5745
This theorem is referenced by:  blrnps  12507  blrn  12508  tgioo  12642
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