MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elrnmpt2 Unicode version

Theorem elrnmpt2 6415
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:   ,   , ,

Proof of Theorem elrnmpt2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 rngop.1 . . . 4
21rnmpt2 6412 . . 3
32eleq2i 2535 . 2
4 elrnmpt2.1 . . . . . 6
5 eleq1 2529 . . . . . 6
64, 5mpbiri 233 . . . . 5
76rexlimivw 2946 . . . 4
87rexlimivw 2946 . . 3
9 eqeq1 2461 . . . 4
1092rexbidv 2975 . . 3
118, 10elab3 3253 . 2
123, 11bitri 249 1
Colors of variables: wff setvar class
Syntax hints:  <->wb 184  =wceq 1395  e.wcel 1818  {cab 2442  E.wrex 2808   cvv 3109  rancrn 5005  e.cmpt2 6298
This theorem is referenced by:  qexALT  11226  lsmelvalx  16660  efgtlen  16744  frgpnabllem1  16877  fmucndlem  20794  mbfimaopnlem  22062  tglnunirn  23935  tpr2rico  27894  mbfmco2  28236  br2base  28240  dya2icobrsiga  28247  dya2iocnrect  28252  dya2iocucvr  28255  sxbrsigalem2  28257  cntotbnd  30292  eldiophb  30690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-br 4453  df-opab 4511  df-cnv 5012  df-dm 5014  df-rn 5015  df-oprab 6300  df-mpt2 6301
  Copyright terms: Public domain W3C validator