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Theorem elrnmpog 5935
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
elrnmpog  |-  ( 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 elrnmpog
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 eqeq1 2164 . . 3  |-  ( z  =  D  ->  (
z  =  C  <->  D  =  C ) )
212rexbidv 2482 . 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 )
43rnmpo 5933 . 2  |-  ran  F  =  { z  |  E. x  e.  A  E. y  e.  B  z  =  C }
52, 4elab2g 2859 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 104    = wceq 1335    e. wcel 2128   E.wrex 2436   ran crn 4589    e. cmpo 5828
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4084  ax-pow 4137  ax-pr 4171
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-br 3968  df-opab 4028  df-cnv 4596  df-dm 4598  df-rn 4599  df-oprab 5830  df-mpo 5831
This theorem is referenced by:  txopn  12735  xmettxlem  12979  xmettx  12980
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