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Theorem dmiun 5911
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 3286 . . . 4 (∃𝑥𝐴𝑧𝑦, 𝑧⟩ ∈ 𝐵 ↔ ∃𝑧𝑥𝐴𝑦, 𝑧⟩ ∈ 𝐵)
2 vex 3479 . . . . . 6 𝑦 ∈ V
32eldm2 5899 . . . . 5 (𝑦 ∈ dom 𝐵 ↔ ∃𝑧𝑦, 𝑧⟩ ∈ 𝐵)
43rexbii 3095 . . . 4 (∃𝑥𝐴 𝑦 ∈ dom 𝐵 ↔ ∃𝑥𝐴𝑧𝑦, 𝑧⟩ ∈ 𝐵)
5 eliun 5000 . . . . 5 (⟨𝑦, 𝑧⟩ ∈ 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴𝑦, 𝑧⟩ ∈ 𝐵)
65exbii 1851 . . . 4 (∃𝑧𝑦, 𝑧⟩ ∈ 𝑥𝐴 𝐵 ↔ ∃𝑧𝑥𝐴𝑦, 𝑧⟩ ∈ 𝐵)
71, 4, 63bitr4ri 304 . . 3 (∃𝑧𝑦, 𝑧⟩ ∈ 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦 ∈ dom 𝐵)
82eldm2 5899 . . 3 (𝑦 ∈ dom 𝑥𝐴 𝐵 ↔ ∃𝑧𝑦, 𝑧⟩ ∈ 𝑥𝐴 𝐵)
9 eliun 5000 . . 3 (𝑦 𝑥𝐴 dom 𝐵 ↔ ∃𝑥𝐴 𝑦 ∈ dom 𝐵)
107, 8, 93bitr4i 303 . 2 (𝑦 ∈ dom 𝑥𝐴 𝐵𝑦 𝑥𝐴 dom 𝐵)
1110eqriv 2730 1 dom 𝑥𝐴 𝐵 = 𝑥𝐴 dom 𝐵
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  wex 1782  wcel 2107  wrex 3071  cop 4633   ciun 4996  dom cdm 5675
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-11 2155  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-iun 4998  df-br 5148  df-dm 5685
This theorem is referenced by:  dprd2d2  19906  gsumpart  32185  esum2d  33029  fmla  34310  iunrelexp0  42386
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