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Theorem fnunirn 6153
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 5678 . . 3  |-  ( F  Fn  I  ->  Fun  F )
2 elunirn 6151 . . 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 5680 . . 3  |-  ( F  Fn  I  ->  dom  F  =  I )
54rexeqdv 3065 . 2  |-  ( F  Fn  I  ->  ( E. x  e.  dom  F  A  e.  ( F `
 x )  <->  E. x  e.  I  A  e.  ( F `  x ) ) )
63, 5bitrd 253 1  |-  ( F  Fn  I  ->  ( A  e.  U. ran  F  <->  E. x  e.  I  A  e.  ( F `  x ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    e. wcel 1767   E.wrex 2815   U.cuni 4245   dom cdm 4999   ran crn 5000   Fun wfun 5582    Fn wfn 5583   ` cfv 5588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5551  df-fun 5590  df-fn 5591  df-fv 5596
This theorem is referenced by:  itunitc  8801  wunex2  9116  mreunirn  14856  arwhoma  15230  filunirn  20146  xmetunirn  20603  abfmpunirn  27190  neibastop2lem  29809  stoweidlem59  31387
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