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Theorem fniunfv 5860
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 5652 . . 3  |-  ( F  Fn  A  ->  ran  F  =  { y  |  E. x  e.  A  y  =  ( F `  x ) } )
21unieqd 3919 . 2  |-  ( F  Fn  A  ->  U. ran  F  =  U. { y  |  E. x  e.  A  y  =  ( F `  x ) } )
3 fvex 5622 . . 3  |-  ( F `
 x )  e. 
_V
43dfiun2 4018 . 2  |-  U_ x  e.  A  ( F `  x )  =  U. { y  |  E. x  e.  A  y  =  ( F `  x ) }
52, 4syl6reqr 2409 1  |-  ( F  Fn  A  ->  U_ x  e.  A  ( F `  x )  =  U. ran  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1642   {cab 2344   E.wrex 2620   U.cuni 3908   U_ciun 3986   ran crn 4772    Fn wfn 5332   ` cfv 5337
This theorem is referenced by:  funiunfv  5861  dffi3  7274  marypha2  7282  jech9.3  7576  hsmexlem5  8146  wuncval2  8459  dprdspan  15361  tgcmp  17234  txcmplem1  17441  txcmplem2  17442  xkococnlem  17459  alexsubALT  17847  bcth3  18857  ovolfioo  18931  ovolficc  18932  voliunlem2  19012  voliunlem3  19013  volsup  19017  uniiccdif  19037  uniioovol  19038  uniiccvol  19039  uniioombllem2  19042  uniioombllem4  19045  volsup2  19064  itg1climres  19173  itg2monolem1  19209  itg2gt0  19219  dftrpred2  24780  volsupnfl  25491  hbt  26657
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pr 4295
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-iota 5301  df-fun 5339  df-fn 5340  df-fv 5345
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