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Theorem fnunirn 5460
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 5048 . . 3  |-  ( F  Fn  I  ->  Fun  F )
2 elunirn 5459 . . 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 5050 . . 3  |-  ( F  Fn  I  ->  dom  F  =  I )
54rexeqdv 2562 . 2  |-  ( F  Fn  I  ->  ( E. x  e.  dom  F  A  e.  ( F `
 x )  <->  E. x  e.  I  A  e.  ( F `  x ) ) )
63, 5bitrd 186 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 103    e. wcel 1434   E.wrex 2354   U.cuni 3622   dom cdm 4392   ran crn 4393   Fun wfun 4947    Fn wfn 4948   ` cfv 4953
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3917  ax-pow 3969  ax-pr 3993
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2613  df-sbc 2826  df-un 2987  df-in 2989  df-ss 2996  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-uni 3623  df-br 3807  df-opab 3861  df-mpt 3862  df-id 4077  df-xp 4398  df-rel 4399  df-cnv 4400  df-co 4401  df-dm 4402  df-rn 4403  df-iota 4918  df-fun 4955  df-fn 4956  df-fv 4961
This theorem is referenced by: (None)
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