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Theorem dmiun 5910
Description: The domain of an indexed union. (Contributed by Mario Carneiro, 26-Apr-2016.)
Assertion
Ref Expression
dmiun dom 𝑥𝐴 𝐵 = 𝑥𝐴 dom 𝐵

Proof of Theorem dmiun
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rexcom4 3281 . . . 4 (∃𝑥𝐴𝑧𝑦, 𝑧⟩ ∈ 𝐵 ↔ ∃𝑧𝑥𝐴𝑦, 𝑧⟩ ∈ 𝐵)
2 vex 3474 . . . . . 6 𝑦 ∈ V
32eldm2 5898 . . . . 5 (𝑦 ∈ dom 𝐵 ↔ ∃𝑧𝑦, 𝑧⟩ ∈ 𝐵)
43rexbii 3090 . . . 4 (∃𝑥𝐴 𝑦 ∈ dom 𝐵 ↔ ∃𝑥𝐴𝑧𝑦, 𝑧⟩ ∈ 𝐵)
5 eliun 4995 . . . . 5 (⟨𝑦, 𝑧⟩ ∈ 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴𝑦, 𝑧⟩ ∈ 𝐵)
65exbii 1843 . . . 4 (∃𝑧𝑦, 𝑧⟩ ∈ 𝑥𝐴 𝐵 ↔ ∃𝑧𝑥𝐴𝑦, 𝑧⟩ ∈ 𝐵)
71, 4, 63bitr4ri 304 . . 3 (∃𝑧𝑦, 𝑧⟩ ∈ 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦 ∈ dom 𝐵)
82eldm2 5898 . . 3 (𝑦 ∈ dom 𝑥𝐴 𝐵 ↔ ∃𝑧𝑦, 𝑧⟩ ∈ 𝑥𝐴 𝐵)
9 eliun 4995 . . 3 (𝑦 𝑥𝐴 dom 𝐵 ↔ ∃𝑥𝐴 𝑦 ∈ dom 𝐵)
107, 8, 93bitr4i 303 . 2 (𝑦 ∈ dom 𝑥𝐴 𝐵𝑦 𝑥𝐴 dom 𝐵)
1110eqriv 2725 1 dom 𝑥𝐴 𝐵 = 𝑥𝐴 dom 𝐵
Colors of variables: wff setvar class
Syntax hints:   = wceq 1534  wex 1774  wcel 2099  wrex 3066  cop 4630   ciun 4991  dom cdm 5672
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-11 2147  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-iun 4993  df-br 5143  df-dm 5682
This theorem is referenced by:  dprd2d2  19994  gsumpart  32763  esum2d  33706  fmla  34985  iunrelexp0  43126
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