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Theorem ovelrn 6451
 Description: A member of an operation's range is a value of the operation. (Contributed by NM, 7-Feb-2007.) (Revised by Mario Carneiro, 30-Jan-2014.)
Assertion
Ref Expression
ovelrn
Distinct variable groups:   ,,   ,,   ,,   ,,

Proof of Theorem ovelrn
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 fnrnov 6448 . . 3
21eleq2d 2527 . 2
3 ovex 6324 . . . . . 6
4 eleq1 2529 . . . . . 6
53, 4mpbiri 233 . . . . 5
65rexlimivw 2946 . . . 4
76rexlimivw 2946 . . 3
8 eqeq1 2461 . . . 4
982rexbidv 2975 . . 3
107, 9elab3 3253 . 2
112, 10syl6bb 261 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  <->wb 184  =wceq 1395  e.wcel 1818  {cab 2442  E.wrex 2808   cvv 3109  X.cxp 5002  rancrn 5005  Fnwfn 5588  (class class class)co 6296 This theorem is referenced by:  efgredlem  16765  efgcpbllemb  16773  gsumval3OLD  16908  gsumval3  16911  lecldbas  19720  blrnps  20911  blrn  20912  qdensere  21277  tgioo  21301  xrge0tsms  21339  ioorf  21982  ioorinv  21985  ioorcl  21986  dyaddisj  22005  dyadmax  22007  mbfid  22043  ismbfd  22047  hhssnv  26180  xrge0tsmsd  27775  iccllyscon  28695  rellyscon  28696  islptre  31625 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-sbc 3328  df-csb 3435  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-uni 4250  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-iota 5556  df-fun 5595  df-fn 5596  df-fv 5601  df-ov 6299
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