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Theorem ovelrn 6238
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 6235 . . 3  |-  ( F  Fn  ( A  X.  B )  ->  ran  F  =  { z  |  E. x  e.  A  E. y  e.  B  z  =  ( x F y ) } )
21eleq2d 2508 . 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 6115 . . . . . 6  |-  ( x F y )  e. 
_V
4 eleq1 2501 . . . . . 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 2835 . . . 4  |-  ( E. y  e.  B  C  =  ( x F y )  ->  C  e.  _V )
76rexlimivw 2835 . . 3  |-  ( E. x  e.  A  E. y  e.  B  C  =  ( x F y )  ->  C  e.  _V )
8 eqeq1 2447 . . . 4  |-  ( z  =  C  ->  (
z  =  ( x F y )  <->  C  =  ( x F y ) ) )
982rexbidv 2756 . . 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 3110 . 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 1364    e. wcel 1761   {cab 2427   E.wrex 2714   _Vcvv 2970    X. cxp 4834   ran crn 4837    Fn wfn 5410  (class class class)co 6090
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pr 4528
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-iota 5378  df-fun 5417  df-fn 5418  df-fv 5423  df-ov 6093
This theorem is referenced by:  efgredlem  16237  efgcpbllemb  16245  gsumval3OLD  16375  gsumval3  16378  lecldbas  18782  blrnps  19942  blrn  19943  qdensere  20308  tgioo  20332  xrge0tsms  20370  ioorf  21012  ioorinv  21015  ioorcl  21016  dyaddisj  21035  dyadmax  21037  mbfid  21073  ismbfd  21077  hhssnv  24600  xrge0tsmsd  26188  iccllyscon  27069  rellyscon  27070
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