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Theorem elrnmpt2g 6185
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 2444 . . 3  |-  ( z  =  D  ->  (
z  =  C  <->  D  =  C ) )
212rexbidv 2750 . 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 6183 . 2  |-  ran  F  =  { z  |  E. x  e.  A  E. y  e.  B  z  =  C }
52, 4elab2g 3086 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 178    = wceq 1653    e. wcel 1726   E.wrex 2708   ran crn 4882    e. cmpt2 6086
This theorem is referenced by:  ordtbas2  17260  txopn  17639  elsx  24553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216  df-opab 4270  df-cnv 4889  df-dm 4891  df-rn 4892  df-oprab 6088  df-mpt2 6089
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