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Theorem elrnmpt2g 6141
Description: Membership in the range of an operation class abstraction. (Contributed by NM, 27-Aug-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)
Hypothesis
Ref Expression
rngop.1  |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )
Assertion
Ref Expression
elrnmpt2g  |-  ( D  e.  V  ->  ( 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)    V( x, y)

Proof of Theorem elrnmpt2g
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 eqeq1 2410 . . 3  |-  ( z  =  D  ->  (
z  =  C  <->  D  =  C ) )
212rexbidv 2709 . 2  |-  ( z  =  D  ->  ( E. x  e.  A  E. y  e.  B  z  =  C  <->  E. x  e.  A  E. y  e.  B  D  =  C ) )
3 rngop.1 . . 3  |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )
43rnmpt2 6139 . 2  |-  ran  F  =  { z  |  E. x  e.  A  E. y  e.  B  z  =  C }
52, 4elab2g 3044 1  |-  ( D  e.  V  ->  ( D  e.  ran  F  <->  E. x  e.  A  E. y  e.  B  D  =  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649    e. wcel 1721   E.wrex 2667   ran crn 4838    e. cmpt2 6042
This theorem is referenced by:  ordtbas2  17209  txopn  17587  elsx  24501
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-cnv 4845  df-dm 4847  df-rn 4848  df-oprab 6044  df-mpt2 6045
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