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Theorem elrnmpo 5884
 Description: Membership in the range of an operation class abstraction. (Contributed by NM, 1-Aug-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)
Hypotheses
Ref Expression
rngop.1
elrnmpo.1
Assertion
Ref Expression
elrnmpo
Distinct variable groups:   ,   ,,
Allowed substitution hints:   ()   (,)   (,)   (,)

Proof of Theorem elrnmpo
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 rngop.1 . . . 4
21rnmpo 5881 . . 3
32eleq2i 2206 . 2
4 elrnmpo.1 . . . . . 6
5 eleq1 2202 . . . . . 6
64, 5mpbiri 167 . . . . 5
76rexlimivw 2545 . . . 4
87rexlimivw 2545 . . 3
9 eqeq1 2146 . . . 4
1092rexbidv 2460 . . 3
118, 10elab3 2836 . 2
123, 11bitri 183 1
 Colors of variables: wff set class Syntax hints:   wb 104   wceq 1331   wcel 1480  cab 2125  wrex 2417  cvv 2686   crn 4540   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-ral 2421  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: (None)
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