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Theorem ovelrn 6254
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 6251 . . 3  |-  ( F  Fn  ( A  X.  B )  ->  ran  F  =  { z  |  E. x  e.  A  E. y  e.  B  z  =  ( x F y ) } )
21eleq2d 2510 . 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 ovex 6131 . . . . . 6  |-  ( x F y )  e. 
_V
4 eleq1 2503 . . . . . 6  |-  ( C  =  ( x F y )  ->  ( C  e.  _V  <->  ( x F y )  e. 
_V ) )
53, 4mpbiri 233 . . . . 5  |-  ( C  =  ( x F y )  ->  C  e.  _V )
65rexlimivw 2852 . . . 4  |-  ( E. y  e.  B  C  =  ( x F y )  ->  C  e.  _V )
76rexlimivw 2852 . . 3  |-  ( E. x  e.  A  E. y  e.  B  C  =  ( x F y )  ->  C  e.  _V )
8 eqeq1 2449 . . . 4  |-  ( z  =  C  ->  (
z  =  ( x F y )  <->  C  =  ( x F y ) ) )
982rexbidv 2773 . . 3  |-  ( 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 ) ) )
107, 9elab3 3128 . 2  |-  ( 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 ) )
112, 10syl6bb 261 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1369    e. wcel 1756   {cab 2429   E.wrex 2731   _Vcvv 2987    X. cxp 4853   ran crn 4856    Fn wfn 5428  (class class class)co 6106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4428  ax-nul 4436  ax-pr 4546
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2735  df-rex 2736  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-nul 3653  df-if 3807  df-sn 3893  df-pr 3895  df-op 3899  df-uni 4107  df-iun 4188  df-br 4308  df-opab 4366  df-mpt 4367  df-id 4651  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-iota 5396  df-fun 5435  df-fn 5436  df-fv 5441  df-ov 6109
This theorem is referenced by:  efgredlem  16259  efgcpbllemb  16267  gsumval3OLD  16397  gsumval3  16400  lecldbas  18838  blrnps  19998  blrn  19999  qdensere  20364  tgioo  20388  xrge0tsms  20426  ioorf  21068  ioorinv  21071  ioorcl  21072  dyaddisj  21091  dyadmax  21093  mbfid  21129  ismbfd  21133  hhssnv  24680  xrge0tsmsd  26268  iccllyscon  27154  rellyscon  27155
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