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Theorem elrnmpt2 5957
Description: Membership in the range of an operation class abstraction. (Contributed by NM, 1-Aug-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)
Hypotheses
Ref Expression
rngop.1  |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )
elrnmpt2.1  |-  C  e. 
_V
Assertion
Ref Expression
elrnmpt2  |-  ( D  e.  ran  F  <->  E. x  e.  A  E. y  e.  B  D  =  C )
Distinct variable groups:    y, A    x, y, D
Allowed substitution hints:    A( x)    B( x, y)    C( x, y)    F( x, y)

Proof of Theorem elrnmpt2
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 rngop.1 . . . 4  |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )
21rnmpt2 5954 . . 3  |-  ran  F  =  { z  |  E. x  e.  A  E. y  e.  B  z  =  C }
32eleq2i 2347 . 2  |-  ( D  e.  ran  F  <->  D  e.  { z  |  E. x  e.  A  E. y  e.  B  z  =  C } )
4 elrnmpt2.1 . . . . . 6  |-  C  e. 
_V
5 eleq1 2343 . . . . . 6  |-  ( D  =  C  ->  ( D  e.  _V  <->  C  e.  _V ) )
64, 5mpbiri 224 . . . . 5  |-  ( D  =  C  ->  D  e.  _V )
76rexlimivw 2663 . . . 4  |-  ( E. y  e.  B  D  =  C  ->  D  e. 
_V )
87rexlimivw 2663 . . 3  |-  ( E. x  e.  A  E. y  e.  B  D  =  C  ->  D  e. 
_V )
9 eqeq1 2289 . . . 4  |-  ( z  =  D  ->  (
z  =  C  <->  D  =  C ) )
1092rexbidv 2586 . . 3  |-  ( z  =  D  ->  ( E. x  e.  A  E. y  e.  B  z  =  C  <->  E. x  e.  A  E. y  e.  B  D  =  C ) )
118, 10elab3 2921 . 2  |-  ( D  e.  { z  |  E. x  e.  A  E. y  e.  B  z  =  C }  <->  E. x  e.  A  E. y  e.  B  D  =  C )
123, 11bitri 240 1  |-  ( D  e.  ran  F  <->  E. x  e.  A  E. y  e.  B  D  =  C )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1623    e. wcel 1684   {cab 2269   E.wrex 2544   _Vcvv 2788   ran crn 4690    e. cmpt2 5860
This theorem is referenced by:  qexALT  10331  lsmelvalx  14951  efgtlen  15035  frgpnabllem1  15161  mbfimaopnlem  19010  tpr2rico  23296  mbfmco2  23570  br2base  23574  dya2iocrrnval  23582  iscst3  25176  cntotbnd  26520  eldiophb  26836
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-cnv 4697  df-dm 4699  df-rn 4700  df-oprab 5862  df-mpt2 5863
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