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Theorem elunirn 5777
Description: Membership in the union of the range of a function. See elunirnALT 5779 for alternate proof. (Contributed by NM, 24-Sep-2006.)
Assertion
Ref Expression
elunirn  |-  ( Fun 
F  ->  ( A  e.  U. ran  F  <->  E. x  e.  dom  F  A  e.  ( F `  x
) ) )
Distinct variable groups:    x, A    x, F

Proof of Theorem elunirn
StepHypRef Expression
1 imadmrn 5024 . . . 4  |-  ( F
" dom  F )  =  ran  F
21unieqi 3837 . . 3  |-  U. ( F " dom  F )  =  U. ran  F
32eleq2i 2347 . 2  |-  ( A  e.  U. ( F
" dom  F )  <->  A  e.  U. ran  F
)
4 eluniima 5776 . 2  |-  ( Fun 
F  ->  ( A  e.  U. ( F " dom  F )  <->  E. x  e.  dom  F  A  e.  ( F `  x
) ) )
53, 4syl5bbr 250 1  |-  ( Fun 
F  ->  ( A  e.  U. ran  F  <->  E. x  e.  dom  F  A  e.  ( F `  x
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    e. wcel 1684   E.wrex 2544   U.cuni 3827   dom cdm 4689   ran crn 4690   "cima 4692   Fun wfun 5249   ` cfv 5255
This theorem is referenced by:  fnunirn  5778  fin23lem30  7968  elunirn2  23215  elunirnmbfm  23558  nsn  25530
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263
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