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Theorem ovelrn 6433
 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 6430 . . 3
21eleq2d 2537 . 2
3 ovex 6307 . . . . . 6
4 eleq1 2539 . . . . . 6
53, 4mpbiri 233 . . . . 5
65rexlimivw 2952 . . . 4
76rexlimivw 2952 . . 3
8 eqeq1 2471 . . . 4
982rexbidv 2980 . . 3
107, 9elab3 3257 . 2
112, 10syl6bb 261 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wceq 1379   wcel 1767  cab 2452  wrex 2815  cvv 3113   cxp 4997   crn 5000   wfn 5581  (class class class)co 6282 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5549  df-fun 5588  df-fn 5589  df-fv 5594  df-ov 6285 This theorem is referenced by:  efgredlem  16561  efgcpbllemb  16569  gsumval3OLD  16699  gsumval3  16702  lecldbas  19486  blrnps  20646  blrn  20647  qdensere  21012  tgioo  21036  xrge0tsms  21074  ioorf  21717  ioorinv  21720  ioorcl  21721  dyaddisj  21740  dyadmax  21742  mbfid  21778  ismbfd  21782  hhssnv  25856  xrge0tsmsd  27438  iccllyscon  28335  rellyscon  28336  islptre  31161
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