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Theorem elrnmpt2 5666
 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
elrnmpt2.1
Assertion
Ref Expression
elrnmpt2
Distinct variable groups:   ,   ,,
Allowed substitution hints:   ()   (,)   (,)   (,)

Proof of Theorem elrnmpt2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 rngop.1 . . . 4
21rnmpt2 5663 . . 3
32eleq2i 2149 . 2
4 elrnmpt2.1 . . . . . 6
5 eleq1 2145 . . . . . 6
64, 5mpbiri 166 . . . . 5
76rexlimivw 2478 . . . 4
87rexlimivw 2478 . . 3
9 eqeq1 2089 . . . 4
1092rexbidv 2396 . . 3
118, 10elab3 2753 . 2
123, 11bitri 182 1
 Colors of variables: wff set class Syntax hints:   wb 103   wceq 1285   wcel 1434  cab 2069  wrex 2354  cvv 2610   crn 4392   cmpt2 5566 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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992 This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-cnv 4399  df-dm 4401  df-rn 4402  df-oprab 5568  df-mpt2 5569 This theorem is referenced by: (None)
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