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Theorem fniunfv 6490
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 𝑥𝐴 (𝐹𝑥) = 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 6229 . . 3 (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)})
21unieqd 4437 . 2 (𝐹 Fn 𝐴 ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)})
3 fvex 6188 . . 3 (𝐹𝑥) ∈ V
43dfiun2 4545 . 2 𝑥𝐴 (𝐹𝑥) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)}
52, 4syl6reqr 2673 1 (𝐹 Fn 𝐴 𝑥𝐴 (𝐹𝑥) = ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1481  {cab 2606  wrex 2910   cuni 4427   ciun 4511  ran crn 5105   Fn wfn 5871  cfv 5876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-iota 5839  df-fun 5878  df-fn 5879  df-fv 5884
This theorem is referenced by:  funiunfv  6491  dffi3  8322  jech9.3  8662  hsmexlem5  9237  wuncval2  9554  dprdspan  18407  tgcmp  21185  txcmplem1  21425  txcmplem2  21426  xkococnlem  21443  alexsubALT  21836  bcth3  23109  ovolfioo  23217  ovolficc  23218  voliunlem2  23300  voliunlem3  23301  volsup  23305  uniiccdif  23327  uniioovol  23328  uniiccvol  23329  uniioombllem2  23332  uniioombllem4  23335  volsup2  23354  itg1climres  23462  itg2monolem1  23498  itg2gt0  23508  sigapildsys  30199  omssubadd  30336  carsgclctunlem3  30356  dftrpred2  31693  volsupnfl  33425  hbt  37519  ovolval4lem1  40626  ovolval5lem3  40631  ovnovollem1  40633  ovnovollem2  40634
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