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Theorem fniunfv 5773
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 5569 . . 3  |-  ( F  Fn  A  ->  ran  F  =  { y  |  E. x  e.  A  y  =  ( F `  x ) } )
21unieqd 3838 . 2  |-  ( F  Fn  A  ->  U. ran  F  =  U. { y  |  E. x  e.  A  y  =  ( F `  x ) } )
3 fvex 5539 . . 3  |-  ( F `
 x )  e. 
_V
43dfiun2 3937 . 2  |-  U_ x  e.  A  ( F `  x )  =  U. { y  |  E. x  e.  A  y  =  ( F `  x ) }
52, 4syl6reqr 2334 1  |-  ( F  Fn  A  ->  U_ x  e.  A  ( F `  x )  =  U. ran  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623   {cab 2269   E.wrex 2544   U.cuni 3827   U_ciun 3905   ran crn 4690    Fn wfn 5250   ` cfv 5255
This theorem is referenced by:  funiunfv  5774  dffi3  7184  marypha2  7192  jech9.3  7486  hsmexlem5  8056  wuncval2  8369  dprdspan  15262  tgcmp  17128  txcmplem1  17335  txcmplem2  17336  xkococnlem  17353  alexsubALT  17745  bcth3  18753  ovolfioo  18827  ovolficc  18828  voliunlem2  18908  voliunlem3  18909  volsup  18913  uniiccdif  18933  uniioovol  18934  uniiccvol  18935  uniioombllem2  18938  uniioombllem4  18941  volsup2  18960  itg1climres  19069  itg2monolem1  19105  itg2gt0  19115  dftrpred2  24222  sallnei  25529  hbt  27334
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263
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