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Theorem fniunfv 5326
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  U_  F `  U. ran  F
Distinct variable groups:   ,   , F

Proof of Theorem fniunfv
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 funfvex 5117 . . . . 5  Fun  F  dom  F  F `  _V
21funfni 4925 . . . 4  F  Fn  F `  _V
32ralrimiva 2370 . . 3  F  Fn  F `  _V
4 dfiun2g 3663 . . 3  F `  _V  U_  F `  U. {  |  F `  }
53, 4syl 14 . 2  F  Fn  U_  F `  U. {  |  F `  }
6 fnrnfv 5145 . . 3  F  Fn  ran  F  {  |  F `  }
76unieqd 3565 . 2  F  Fn  U. ran  F  U. {  |  F `  }
85, 7eqtr4d 2057 1  F  Fn  U_  F `  U. ran  F
Colors of variables: wff set class
Syntax hints:   wi 4   wceq 1228   wcel 1374   {cab 2008  wral 2284  wrex 2285   _Vcvv 2535   U.cuni 3554   U_ciun 3631   ran crn 4273    Fn wfn 4824   ` cfv 4829
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-iun 3633  df-br 3739  df-opab 3793  df-mpt 3794  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-iota 4794  df-fun 4831  df-fn 4832  df-fv 4837
This theorem is referenced by:  funiunfvdm  5327
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