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Theorem fnunirn 5958
Description: Membership in a union of some function-defined family of sets. (Contributed by Stefan O'Rear, 30-Jan-2015.)
Assertion
Ref Expression
fnunirn  |-  ( F  Fn  I  ->  ( A  e.  U. ran  F  <->  E. x  e.  I  A  e.  ( F `  x ) ) )
Distinct variable groups:    x, A    x, I    x, F

Proof of Theorem fnunirn
StepHypRef Expression
1 fnfun 5501 . . 3  |-  ( F  Fn  I  ->  Fun  F )
2 elunirn 5957 . . 3  |-  ( Fun 
F  ->  ( A  e.  U. ran  F  <->  E. x  e.  dom  F  A  e.  ( F `  x
) ) )
31, 2syl 16 . 2  |-  ( F  Fn  I  ->  ( A  e.  U. ran  F  <->  E. x  e.  dom  F  A  e.  ( F `  x ) ) )
4 fndm 5503 . . 3  |-  ( F  Fn  I  ->  dom  F  =  I )
54rexeqdv 2871 . 2  |-  ( F  Fn  I  ->  ( E. x  e.  dom  F  A  e.  ( F `
 x )  <->  E. x  e.  I  A  e.  ( F `  x ) ) )
63, 5bitrd 245 1  |-  ( F  Fn  I  ->  ( A  e.  U. ran  F  <->  E. x  e.  I  A  e.  ( F `  x ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    e. wcel 1721   E.wrex 2667   U.cuni 3975   dom cdm 4837   ran crn 4838   Fun wfun 5407    Fn wfn 5408   ` cfv 5413
This theorem is referenced by:  itunitc  8257  wunex2  8569  mreunirn  13781  arwhoma  14155  filunirn  17867  xmetunirn  18320  abfmpunirn  24017  neibastop2lem  26279  stoweidlem59  27675
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-fv 5421
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