Theorem fniunfv 6409

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		fnrnfv 6152	. . 3 ⊢ (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)})
2	1	unieqd 4382	. 2 ⊢ (𝐹 Fn 𝐴 → ∪ ran 𝐹 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)})
3		fvex 6113	. . 3 ⊢ (𝐹‘𝑥) ∈ V
4	3	dfiun2 4490	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)}
5	2, 4	syl6reqr 2663	1 ⊢ (𝐹 Fn 𝐴 → ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∪ ran 𝐹)

Syntax hints:   → wi 4   = wceq 1475  {cab 2596  ∃wrex 2897  ∪ cuni 4372  ∪ ciun 4455  ran crn 5039   Fn wfn 5799  ‘cfv 5804

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812

This theorem is referenced by:  funiunfv  6410  dffi3  8220  jech9.3  8560  hsmexlem5  9135  wuncval2  9448  dprdspan  18249  tgcmp  21014  txcmplem1  21254  txcmplem2  21255  xkococnlem  21272  alexsubALT  21665  bcth3  22936  ovolfioo  23043  ovolficc  23044  voliunlem2  23126  voliunlem3  23127  volsup  23131  uniiccdif  23152  uniioovol  23153  uniiccvol  23154  uniioombllem2  23157  uniioombllem4  23160  volsup2  23179  itg1climres  23287  itg2monolem1  23323  itg2gt0  23333  sigapildsys  29552  omssubadd  29689  carsgclctunlem3  29709  dftrpred2  30963  volsupnfl  32624  hbt  36719  ovolval4lem1  39539  ovolval5lem3  39544  ovnovollem1  39546  ovnovollem2  39547