Theorem fniunfv 7267

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 → ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∪ ran 𝐹)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐹

Proof of Theorem fniunfv

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvex 6919	. . 3 ⊢ (𝐹‘𝑥) ∈ V
2	1	dfiun2 5033	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)}
3		fnrnfv 6968	. . 3 ⊢ (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)})
4	3	unieqd 4920	. 2 ⊢ (𝐹 Fn 𝐴 → ∪ ran 𝐹 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)})
5	2, 4	eqtr4id 2796	1 ⊢ (𝐹 Fn 𝐴 → ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∪ ran 𝐹)

Syntax hints:   → wi 4   = wceq 1540  {cab 2714  ∃wrex 3070  ∪ cuni 4907  ∪ ciun 4991  ran crn 5686   Fn wfn 6556  ‘cfv 6561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569

This theorem is referenced by:  funiunfv  7268  dffi3  9471  jech9.3  9854  hsmexlem5  10470  wuncval2  10787  dprdspan  20047  tgcmp  23409  txcmplem1  23649  txcmplem2  23650  xkococnlem  23667  alexsubALT  24059  bcth3  25365  ovolfioo  25502  ovolficc  25503  voliunlem2  25586  voliunlem3  25587  volsup  25591  uniiccdif  25613  uniioovol  25614  uniiccvol  25615  uniioombllem2  25618  uniioombllem4  25621  volsup2  25640  itg1climres  25749  itg2monolem1  25785  itg2gt0  25795  sigapildsys  34163  omssubadd  34302  carsgclctunlem3  34322  pibt2  37418  volsupnfl  37672  hbt  43142  ovolval4lem1  46664  ovolval5lem3  46669  ovnovollem1  46671  ovnovollem2  46672