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Theorem elrnmpog 5883
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 2146 . . 3  |-  ( z  =  D  ->  (
z  =  C  <->  D  =  C ) )
212rexbidv 2460 . 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 5881 . 2  |-  ran  F  =  { z  |  E. x  e.  A  E. y  e.  B  z  =  C }
52, 4elab2g 2831 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 1331    e. wcel 1480   E.wrex 2417   ran crn 4540    e. cmpo 5776
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-cnv 4547  df-dm 4549  df-rn 4550  df-oprab 5778  df-mpo 5779
This theorem is referenced by:  txopn  12443  xmettxlem  12687  xmettx  12688
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