Theorem dmiun 5893

Description: The domain of an indexed union. (Contributed by Mario Carneiro, 26-Apr-2016.)

Assertion

Ref	Expression
dmiun	⊢ dom ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 dom 𝐵

Proof of Theorem dmiun

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rexcom4 3269	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑧⟨𝑦, 𝑧⟩ ∈ 𝐵 ↔ ∃𝑧∃𝑥 ∈ 𝐴 ⟨𝑦, 𝑧⟩ ∈ 𝐵)
2		vex 3463	. . . . . 6 ⊢ 𝑦 ∈ V
3	2	eldm2 5881	. . . . 5 ⊢ (𝑦 ∈ dom 𝐵 ↔ ∃𝑧⟨𝑦, 𝑧⟩ ∈ 𝐵)
4	3	rexbii 3083	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ dom 𝐵 ↔ ∃𝑥 ∈ 𝐴 ∃𝑧⟨𝑦, 𝑧⟩ ∈ 𝐵)
5		eliun 4971	. . . . 5 ⊢ (⟨𝑦, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 ⟨𝑦, 𝑧⟩ ∈ 𝐵)
6	5	exbii 1848	. . . 4 ⊢ (∃𝑧⟨𝑦, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑧∃𝑥 ∈ 𝐴 ⟨𝑦, 𝑧⟩ ∈ 𝐵)
7	1, 4, 6	3bitr4ri 304	. . 3 ⊢ (∃𝑧⟨𝑦, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ dom 𝐵)
8	2	eldm2 5881	. . 3 ⊢ (𝑦 ∈ dom ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑧⟨𝑦, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝐴 𝐵)
9		eliun 4971	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 dom 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ dom 𝐵)
10	7, 8, 9	3bitr4i 303	. 2 ⊢ (𝑦 ∈ dom ∪ 𝑥 ∈ 𝐴 𝐵 ↔ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 dom 𝐵)
11	10	eqriv 2732	1 ⊢ dom ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 dom 𝐵

Syntax hints:   = wceq 1540  ∃wex 1779   ∈ wcel 2108  ∃wrex 3060  ⟨cop 4607  ∪ ciun 4967  dom cdm 5654

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2157  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-iun 4969  df-br 5120  df-dm 5664

This theorem is referenced by:  dprd2d2  20025  dmdju  32571  gsumpart  32997  esum2d  34070  fmla  35349  iunrelexp0  43673