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Theorem fniunfv 6140
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 5900 . . 3  |-  ( F  Fn  A  ->  ran  F  =  { y  |  E. x  e.  A  y  =  ( F `  x ) } )
21unieqd 4240 . 2  |-  ( F  Fn  A  ->  U. ran  F  =  U. { y  |  E. x  e.  A  y  =  ( F `  x ) } )
3 fvex 5862 . . 3  |-  ( F `
 x )  e. 
_V
43dfiun2 4345 . 2  |-  U_ x  e.  A  ( F `  x )  =  U. { y  |  E. x  e.  A  y  =  ( F `  x ) }
52, 4syl6reqr 2501 1  |-  ( F  Fn  A  ->  U_ x  e.  A  ( F `  x )  =  U. ran  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1381   {cab 2426   E.wrex 2792   U.cuni 4230   U_ciun 4311   ran crn 4986    Fn wfn 5569   ` cfv 5574
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pr 4672
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-sbc 3312  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-iota 5537  df-fun 5576  df-fn 5577  df-fv 5582
This theorem is referenced by:  funiunfv  6141  dffi3  7889  jech9.3  8230  hsmexlem5  8808  wuncval2  9123  dprdspan  16942  tgcmp  19767  txcmplem1  20008  txcmplem2  20009  xkococnlem  20026  alexsubALT  20417  bcth3  21636  ovolfioo  21745  ovolficc  21746  voliunlem2  21827  voliunlem3  21828  volsup  21832  uniiccdif  21853  uniioovol  21854  uniiccvol  21855  uniioombllem2  21858  uniioombllem4  21861  volsup2  21880  itg1climres  21987  itg2monolem1  22023  itg2gt0  22033  dftrpred2  29270  volsupnfl  30027  hbt  31047
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