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Theorem fnunirn 5560
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 5124 . . 3  |-  ( F  Fn  I  ->  Fun  F )
2 elunirn 5559 . . 3  |-  ( Fun 
F  ->  ( A  e.  U. ran  F  <->  E. x  e.  dom  F  A  e.  ( F `  x
) ) )
31, 2syl 14 . 2  |-  ( F  Fn  I  ->  ( A  e.  U. ran  F  <->  E. x  e.  dom  F  A  e.  ( F `  x ) ) )
4 fndm 5126 . . 3  |-  ( F  Fn  I  ->  dom  F  =  I )
54rexeqdv 2570 . 2  |-  ( F  Fn  I  ->  ( E. x  e.  dom  F  A  e.  ( F `
 x )  <->  E. x  e.  I  A  e.  ( F `  x ) ) )
63, 5bitrd 187 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 104    e. wcel 1439   E.wrex 2361   U.cuni 3659   dom cdm 4452   ran crn 4453   Fun wfun 5022    Fn wfn 5023   ` cfv 5028
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-sbc 2842  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-mpt 3907  df-id 4129  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-iota 4993  df-fun 5030  df-fn 5031  df-fv 5036
This theorem is referenced by: (None)
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