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Theorem elunirn 5937
Description: Membership in the union of the range of a function. See elunirnALT 5939 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 5155 . . . 4  |-  ( F
" dom  F )  =  ran  F
21unieqi 3967 . . 3  |-  U. ( F " dom  F )  =  U. ran  F
32eleq2i 2451 . 2  |-  ( A  e.  U. ( F
" dom  F )  <->  A  e.  U. ran  F
)
4 eluniima 5936 . 2  |-  ( Fun 
F  ->  ( A  e.  U. ( F " dom  F )  <->  E. x  e.  dom  F  A  e.  ( F `  x
) ) )
53, 4syl5bbr 251 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 177    e. wcel 1717   E.wrex 2650   U.cuni 3957   dom cdm 4818   ran crn 4819   "cima 4821   Fun wfun 5388   ` cfv 5394
This theorem is referenced by:  fnunirn  5938  fin23lem30  8155  ustn0  18171  elrnust  18175  ustbas  18178  metuval  18469  elunirn2  23905  elunirnmbfm  24397
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-fv 5402
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