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Theorem fniunfv 6138
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 fnrnfv 5905 . . 3  |-  ( F  Fn  A  ->  ran  F  =  { y  |  E. x  e.  A  y  =  ( F `  x ) } )
21unieqd 4248 . 2  |-  ( F  Fn  A  ->  U. ran  F  =  U. { y  |  E. x  e.  A  y  =  ( F `  x ) } )
3 fvex 5867 . . 3  |-  ( F `
 x )  e. 
_V
43dfiun2 4352 . 2  |-  U_ x  e.  A  ( F `  x )  =  U. { y  |  E. x  e.  A  y  =  ( F `  x ) }
52, 4syl6reqr 2520 1  |-  ( F  Fn  A  ->  U_ x  e.  A  ( F `  x )  =  U. ran  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1374   {cab 2445   E.wrex 2808   U.cuni 4238   U_ciun 4318   ran crn 4993    Fn wfn 5574   ` cfv 5579
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-iota 5542  df-fun 5581  df-fn 5582  df-fv 5587
This theorem is referenced by:  funiunfv  6139  dffi3  7880  jech9.3  8221  hsmexlem5  8799  wuncval2  9114  dprdspan  16857  tgcmp  19660  txcmplem1  19870  txcmplem2  19871  xkococnlem  19888  alexsubALT  20279  bcth3  21498  ovolfioo  21607  ovolficc  21608  voliunlem2  21689  voliunlem3  21690  volsup  21694  uniiccdif  21715  uniioovol  21716  uniiccvol  21717  uniioombllem2  21720  uniioombllem4  21723  volsup2  21742  itg1climres  21849  itg2monolem1  21885  itg2gt0  21895  dftrpred2  28729  volsupnfl  29487  hbt  30536
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