Theorem fniunfv 7008

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 6727	. . 3 ⊢ (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)})
2	1	unieqd 4854	. 2 ⊢ (𝐹 Fn 𝐴 → ∪ ran 𝐹 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)})
3		fvex 6685	. . 3 ⊢ (𝐹‘𝑥) ∈ V
4	3	dfiun2 4960	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)}
5	2, 4	syl6reqr 2877	1 ⊢ (𝐹 Fn 𝐴 → ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∪ ran 𝐹)

Syntax hints:   → wi 4   = wceq 1537  {cab 2801  ∃wrex 3141  ∪ cuni 4840  ∪ ciun 4921  ran crn 5558   Fn wfn 6352  ‘cfv 6357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-iota 6316  df-fun 6359  df-fn 6360  df-fv 6365

This theorem is referenced by:  funiunfv  7009  dffi3  8897  jech9.3  9245  hsmexlem5  9854  wuncval2  10171  dprdspan  19151  tgcmp  22011  txcmplem1  22251  txcmplem2  22252  xkococnlem  22269  alexsubALT  22661  bcth3  23936  ovolfioo  24070  ovolficc  24071  voliunlem2  24154  voliunlem3  24155  volsup  24159  uniiccdif  24181  uniioovol  24182  uniiccvol  24183  uniioombllem2  24186  uniioombllem4  24189  volsup2  24208  itg1climres  24317  itg2monolem1  24353  itg2gt0  24363  sigapildsys  31423  omssubadd  31560  carsgclctunlem3  31580  dftrpred2  33060  pibt2  34700  volsupnfl  34939  hbt  39737  ovolval4lem1  42938  ovolval5lem3  42943  ovnovollem1  42945  ovnovollem2  42946