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Theorem elunirn 5990
Description: Membership in the union of the range of a function. See elunirnALT 5992 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 5207 . . . 4  |-  ( F
" dom  F )  =  ran  F
21unieqi 4017 . . 3  |-  U. ( F " dom  F )  =  U. ran  F
32eleq2i 2499 . 2  |-  ( A  e.  U. ( F
" dom  F )  <->  A  e.  U. ran  F
)
4 eluniima 5989 . 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 1725   E.wrex 2698   U.cuni 4007   dom cdm 4870   ran crn 4871   "cima 4873   Fun wfun 5440   ` cfv 5446
This theorem is referenced by:  fnunirn  5991  fin23lem30  8214  ustn0  18242  elrnust  18246  ustbas  18249  metuvalOLD  18571  metuval  18572  elunirn2  24055  metidval  24277  pstmval  24282  elunirnmbfm  24595
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-fv 5454
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