Theorem dmiun 5822

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 3233	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑧⟨𝑦, 𝑧⟩ ∈ 𝐵 ↔ ∃𝑧∃𝑥 ∈ 𝐴 ⟨𝑦, 𝑧⟩ ∈ 𝐵)
2		vex 3436	. . . . . 6 ⊢ 𝑦 ∈ V
3	2	eldm2 5810	. . . . 5 ⊢ (𝑦 ∈ dom 𝐵 ↔ ∃𝑧⟨𝑦, 𝑧⟩ ∈ 𝐵)
4	3	rexbii 3181	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ dom 𝐵 ↔ ∃𝑥 ∈ 𝐴 ∃𝑧⟨𝑦, 𝑧⟩ ∈ 𝐵)
5		eliun 4928	. . . . 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 5810	. . 3 ⊢ (𝑦 ∈ dom ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑧⟨𝑦, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝐴 𝐵)
9		eliun 4928	. . 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 2735	1 ⊢ dom ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 dom 𝐵

Syntax hints:   = wceq 1539  ∃wex 1782   ∈ wcel 2106  ∃wrex 3065  ⟨cop 4567  ∪ ciun 4924  dom cdm 5589

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-iun 4926  df-br 5075  df-dm 5599

This theorem is referenced by:  dprd2d2  19647  gsumpart  31315  esum2d  32061  fmla  33343  iunrelexp0  41310