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Theorem fnunirn 5865
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 5423 . . 3  |-  ( F  Fn  I  ->  Fun  F )
2 elunirn 5864 . . 3  |-  ( Fun 
F  ->  ( A  e.  U. ran  F  <->  E. x  e.  dom  F  A  e.  ( F `  x
) ) )
31, 2syl 15 . 2  |-  ( F  Fn  I  ->  ( A  e.  U. ran  F  <->  E. x  e.  dom  F  A  e.  ( F `  x ) ) )
4 fndm 5425 . . 3  |-  ( F  Fn  I  ->  dom  F  =  I )
54rexeqdv 2819 . 2  |-  ( F  Fn  I  ->  ( E. x  e.  dom  F  A  e.  ( F `
 x )  <->  E. x  e.  I  A  e.  ( F `  x ) ) )
63, 5bitrd 244 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 176    e. wcel 1710   E.wrex 2620   U.cuni 3908   dom cdm 4771   ran crn 4772   Fun wfun 5331    Fn wfn 5332   ` cfv 5337
This theorem is referenced by:  itunitc  8137  wunex2  8450  mreunirn  13602  arwhoma  13976  filunirn  17679  xmetunirn  18004  abfmpunirn  23267  neibastop2lem  25633  stoweidlem59  27131
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-fv 5345
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