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Theorem fniunfv 6985
 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 fvex 6659 . . 3 (𝐹𝑥) ∈ V
21dfiun2 4921 . 2 𝑥𝐴 (𝐹𝑥) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)}
3 fnrnfv 6701 . . 3 (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)})
43unieqd 4815 . 2 (𝐹 Fn 𝐴 ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)})
52, 4eqtr4id 2852 1 (𝐹 Fn 𝐴 𝑥𝐴 (𝐹𝑥) = ran 𝐹)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538  {cab 2776  ∃wrex 3107  ∪ cuni 4801  ∪ ciun 4882  ran crn 5521   Fn wfn 6320  ‘cfv 6325 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pr 5296 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-iota 6284  df-fun 6327  df-fn 6328  df-fv 6333 This theorem is referenced by:  funiunfv  6986  dffi3  8882  jech9.3  9230  hsmexlem5  9844  wuncval2  10161  dprdspan  19146  tgcmp  22016  txcmplem1  22256  txcmplem2  22257  xkococnlem  22274  alexsubALT  22666  bcth3  23945  ovolfioo  24081  ovolficc  24082  voliunlem2  24165  voliunlem3  24166  volsup  24170  uniiccdif  24192  uniioovol  24193  uniiccvol  24194  uniioombllem2  24197  uniioombllem4  24200  volsup2  24219  itg1climres  24328  itg2monolem1  24364  itg2gt0  24374  sigapildsys  31546  omssubadd  31683  carsgclctunlem3  31703  dftrpred2  33186  pibt2  34853  volsupnfl  35121  hbt  40117  ovolval4lem1  43331  ovolval5lem3  43336  ovnovollem1  43338  ovnovollem2  43339
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