Theorem fniunfv 7271

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 6918	. . 3 ⊢ (𝐹‘𝑥) ∈ V
2	1	dfiun2 4993	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)}
3		fnrnfv 6964	. . 3 ⊢ (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)})
4	3	unieqd 4883	. 2 ⊢ (𝐹 Fn 𝐴 → ∪ ran 𝐹 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)})
5	2, 4	eqtr4id 2817	1 ⊢ (𝐹 Fn 𝐴 → ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∪ ran 𝐹)

This proof depends on syntax axioms:   → wi 4   = wceq 1570  {cab 2741  ∃wrex 3090  ∪ cuni 4870  ∪ ciun 4954  ran crn 5656   Fn wfn 6550  ‘cfv 6555

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-iota 6509  df-fun 6557  df-fn 6558  df-fv 6563

This theorem is used by:  funiunfv  7272  dffi3  9447  jech9.3OLD  9847  hsmexlem5  10532  wuncval2  10856  dprdspan  20270  tgcmp  23746  txcmplem1  23987  txcmplem2  23988  xkococnlem  24005  alexsubALT  24397  bcth3  25679  ovolfioo  25815  ovolficc  25816  voliunlem2  25899  voliunlem3  25900  volsup  25904  uniiccdif  25926  uniioovol  25927  uniiccvol  25928  uniioombllem2  25931  uniioombllem4  25934  volsup2  25953  itg1climres  26062  itg2monolem1  26098  itg2gt0  26108  sigapildsys  34862  omssubadd  34999  carsgclctunlem3  35019  pibt2  38531  volsupnfl  38774  hbt  44322  ovolval4lem1  47862  ovolval5lem3  47867  ovnovollem1  47869  ovnovollem2  47870