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Theorem fniunfv 7193
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 6855 . . 3 (𝐹𝑥) ∈ V
21dfiun2 4993 . 2 𝑥𝐴 (𝐹𝑥) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)}
3 fnrnfv 6902 . . 3 (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)})
43unieqd 4879 . 2 (𝐹 Fn 𝐴 ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)})
52, 4eqtr4id 2795 1 (𝐹 Fn 𝐴 𝑥𝐴 (𝐹𝑥) = ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  {cab 2713  wrex 3073   cuni 4865   ciun 4954  ran crn 5634   Fn wfn 6491  cfv 6496
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-iota 6448  df-fun 6498  df-fn 6499  df-fv 6504
This theorem is referenced by:  funiunfv  7194  dffi3  9366  jech9.3  9749  hsmexlem5  10365  wuncval2  10682  dprdspan  19804  tgcmp  22750  txcmplem1  22990  txcmplem2  22991  xkococnlem  23008  alexsubALT  23400  bcth3  24693  ovolfioo  24829  ovolficc  24830  voliunlem2  24913  voliunlem3  24914  volsup  24918  uniiccdif  24940  uniioovol  24941  uniiccvol  24942  uniioombllem2  24945  uniioombllem4  24948  volsup2  24967  itg1climres  25077  itg2monolem1  25113  itg2gt0  25123  sigapildsys  32701  omssubadd  32840  carsgclctunlem3  32860  pibt2  35878  volsupnfl  36113  hbt  41434  ovolval4lem1  44861  ovolval5lem3  44866  ovnovollem1  44868  ovnovollem2  44869
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