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Theorem fniunfv 5401
Description: The indexed union of a function's values is the union of its range. Compare Definition 5.4 of [Monk1] p. 50. (Contributed by NM, 27-Sep-2004.)
Assertion
Ref Expression
fniunfv  |-  ( F  Fn  A  ->  U_ x  e.  A  ( F `  x )  =  U. ran  F )
Distinct variable groups:    x, A    x, F

Proof of Theorem fniunfv
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 funfvex 5192 . . . . 5  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( F `  x
)  e.  _V )
21funfni 4999 . . . 4  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( F `  x
)  e.  _V )
32ralrimiva 2392 . . 3  |-  ( F  Fn  A  ->  A. x  e.  A  ( F `  x )  e.  _V )
4 dfiun2g 3689 . . 3  |-  ( A. x  e.  A  ( F `  x )  e.  _V  ->  U_ x  e.  A  ( F `  x )  =  U. { y  |  E. x  e.  A  y  =  ( F `  x ) } )
53, 4syl 14 . 2  |-  ( F  Fn  A  ->  U_ x  e.  A  ( F `  x )  =  U. { y  |  E. x  e.  A  y  =  ( F `  x ) } )
6 fnrnfv 5220 . . 3  |-  ( F  Fn  A  ->  ran  F  =  { y  |  E. x  e.  A  y  =  ( F `  x ) } )
76unieqd 3591 . 2  |-  ( F  Fn  A  ->  U. ran  F  =  U. { y  |  E. x  e.  A  y  =  ( F `  x ) } )
85, 7eqtr4d 2075 1  |-  ( F  Fn  A  ->  U_ x  e.  A  ( F `  x )  =  U. ran  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1243    e. wcel 1393   {cab 2026   A.wral 2306   E.wrex 2307   _Vcvv 2557   U.cuni 3580   U_ciun 3657   ran crn 4346    Fn wfn 4897   ` cfv 4902
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-iota 4867  df-fun 4904  df-fn 4905  df-fv 4910
This theorem is referenced by:  funiunfvdm  5402
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