Theorem dmiun 5857

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 3262	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑧⟨𝑦, 𝑧⟩ ∈ 𝐵 ↔ ∃𝑧∃𝑥 ∈ 𝐴 ⟨𝑦, 𝑧⟩ ∈ 𝐵)
2		vex 3431	. . . . . 6 ⊢ 𝑦 ∈ V
3	2	eldm2 5845	. . . . 5 ⊢ (𝑦 ∈ dom 𝐵 ↔ ∃𝑧⟨𝑦, 𝑧⟩ ∈ 𝐵)
4	3	rexbii 3082	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ dom 𝐵 ↔ ∃𝑥 ∈ 𝐴 ∃𝑧⟨𝑦, 𝑧⟩ ∈ 𝐵)
5		eliun 4927	. . . . 5 ⊢ (⟨𝑦, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 ⟨𝑦, 𝑧⟩ ∈ 𝐵)
6	5	exbii 1850	. . . 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 5845	. . 3 ⊢ (𝑦 ∈ dom ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑧⟨𝑦, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝐴 𝐵)
9		eliun 4927	. . 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 1542  ∃wex 1781   ∈ wcel 2114  ∃wrex 3059  ⟨cop 4563  ∪ ciun 4923  dom cdm 5620

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-11 2163  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-rex 3060  df-rab 3388  df-v 3429  df-dif 3888  df-un 3890  df-ss 3902  df-nul 4264  df-if 4457  df-sn 4558  df-pr 4560  df-op 4564  df-iun 4925  df-br 5075  df-dm 5630

This theorem is referenced by:  dprd2d2  20010  dmdju  32708  gsumpart  33112  esum2d  34225  fmla  35551  iunrelexp0  44117