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Theorem fniunfv 5959
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 5733 . . 3  |-  ( F  Fn  A  ->  ran  F  =  { y  |  E. x  e.  A  y  =  ( F `  x ) } )
21unieqd 4096 . 2  |-  ( F  Fn  A  ->  U. ran  F  =  U. { y  |  E. x  e.  A  y  =  ( F `  x ) } )
3 fvex 5696 . . 3  |-  ( F `
 x )  e. 
_V
43dfiun2 4199 . 2  |-  U_ x  e.  A  ( F `  x )  =  U. { y  |  E. x  e.  A  y  =  ( F `  x ) }
52, 4syl6reqr 2489 1  |-  ( F  Fn  A  ->  U_ x  e.  A  ( F `  x )  =  U. ran  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369   {cab 2424   E.wrex 2711   U.cuni 4086   U_ciun 4166   ran crn 4836    Fn wfn 5408   ` cfv 5413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pr 4526
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-iota 5376  df-fun 5415  df-fn 5416  df-fv 5421
This theorem is referenced by:  funiunfv  5960  dffi3  7673  jech9.3  8013  hsmexlem5  8591  wuncval2  8906  dprdspan  16512  tgcmp  18979  txcmplem1  19189  txcmplem2  19190  xkococnlem  19207  alexsubALT  19598  bcth3  20817  ovolfioo  20926  ovolficc  20927  voliunlem2  21007  voliunlem3  21008  volsup  21012  uniiccdif  21033  uniioovol  21034  uniiccvol  21035  uniioombllem2  21038  uniioombllem4  21041  volsup2  21060  itg1climres  21167  itg2monolem1  21203  itg2gt0  21213  dftrpred2  27634  volsupnfl  28389  hbt  29439
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