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Theorem fniunfv 5734
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
Dummy variable  y is distinct from all other variables.

Proof of Theorem fniunfv
StepHypRef Expression
1 fnrnfv 5530 . . 3  |-  ( F  Fn  A  ->  ran  F  =  { y  |  E. x  e.  A  y  =  ( F `  x ) } )
21unieqd 3839 . 2  |-  ( F  Fn  A  ->  U. ran  F  =  U. { y  |  E. x  e.  A  y  =  ( F `  x ) } )
3 fvex 5499 . . 3  |-  ( F `
 x )  e. 
_V
43dfiun2 3938 . 2  |-  U_ x  e.  A  ( F `  x )  =  U. { y  |  E. x  e.  A  y  =  ( F `  x ) }
52, 4syl6reqr 2335 1  |-  ( F  Fn  A  ->  U_ x  e.  A  ( F `  x )  =  U. ran  F )
Colors of variables: wff set class
Syntax hints:    -> wi 6    = wceq 1624   {cab 2270   E.wrex 2545   U.cuni 3828   U_ciun 3906   ran crn 4689    Fn wfn 5216   ` cfv 5221
This theorem is referenced by:  funiunfv  5735  dffi3  7179  marypha2  7187  jech9.3  7481  hsmexlem5  8051  wuncval2  8364  dprdspan  15256  tgcmp  17122  txcmplem1  17329  txcmplem2  17330  xkococnlem  17347  alexsubALT  17739  bcth3  18747  ovolfioo  18821  ovolficc  18822  voliunlem2  18902  voliunlem3  18903  volsup  18907  uniiccdif  18927  uniioovol  18928  uniiccvol  18929  uniioombllem2  18932  uniioombllem4  18935  volsup2  18954  itg1climres  19063  itg2monolem1  19099  itg2gt0  19109  dftrpred2  23623  sallnei  24928  hbt  26733
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-fv 5229
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