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Theorem elunirn 5793
Description: Membership in the union of the range of a function. See elunirnALT 5795 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 5040 . . . 4  |-  ( F
" dom  F )  =  ran  F
21unieqi 3853 . . 3  |-  U. ( F " dom  F )  =  U. ran  F
32eleq2i 2360 . 2  |-  ( A  e.  U. ( F
" dom  F )  <->  A  e.  U. ran  F
)
4 eluniima 5792 . 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 1696   E.wrex 2557   U.cuni 3843   dom cdm 4705   ran crn 4706   "cima 4708   Fun wfun 5265   ` cfv 5271
This theorem is referenced by:  fnunirn  5794  fin23lem30  7984  elunirn2  23230  elunirnmbfm  23573  nsn  25633
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-fv 5279
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