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Theorem elrnmpt2g 5757
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 2094 . . 3  |-  ( z  =  D  ->  (
z  =  C  <->  D  =  C ) )
212rexbidv 2403 . 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 5755 . 2  |-  ran  F  =  { z  |  E. x  e.  A  E. y  e.  B  z  =  C }
52, 4elab2g 2762 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 103    = wceq 1289    e. wcel 1438   E.wrex 2360   ran crn 4439    |-> cmpt2 5654
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-cnv 4446  df-dm 4448  df-rn 4449  df-oprab 5656  df-mpt2 5657
This theorem is referenced by: (None)
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