Theorem dmuni 5861

Description: The domain of a union. Part of Exercise 8 of [Enderton] p. 41. (Contributed by NM, 3-Feb-2004.)

Assertion

Ref	Expression
dmuni	⊢ dom ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 dom 𝑥

Distinct variable group:   𝑥,𝐴

Proof of Theorem dmuni

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

Step	Hyp	Ref	Expression
1		excom 2167	. . . . 5 ⊢ (∃𝑧∃𝑥(⟨𝑦, 𝑧⟩ ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) ↔ ∃𝑥∃𝑧(⟨𝑦, 𝑧⟩ ∈ 𝑥 ∧ 𝑥 ∈ 𝐴))
2		ancom 460	. . . . . . 7 ⊢ ((∃𝑧⟨𝑦, 𝑧⟩ ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑧⟨𝑦, 𝑧⟩ ∈ 𝑥))
3		19.41v 1950	. . . . . . 7 ⊢ (∃𝑧(⟨𝑦, 𝑧⟩ ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) ↔ (∃𝑧⟨𝑦, 𝑧⟩ ∈ 𝑥 ∧ 𝑥 ∈ 𝐴))
4		vex 3442	. . . . . . . . 9 ⊢ 𝑦 ∈ V
5	4	eldm2 5848	. . . . . . . 8 ⊢ (𝑦 ∈ dom 𝑥 ↔ ∃𝑧⟨𝑦, 𝑧⟩ ∈ 𝑥)
6	5	anbi2i 623	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ dom 𝑥) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑧⟨𝑦, 𝑧⟩ ∈ 𝑥))
7	2, 3, 6	3bitr4i 303	. . . . . 6 ⊢ (∃𝑧(⟨𝑦, 𝑧⟩ ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ dom 𝑥))
8	7	exbii 1849	. . . . 5 ⊢ (∃𝑥∃𝑧(⟨𝑦, 𝑧⟩ ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ dom 𝑥))
9	1, 8	bitri 275	. . . 4 ⊢ (∃𝑧∃𝑥(⟨𝑦, 𝑧⟩ ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ dom 𝑥))
10		eluni 4863	. . . . 5 ⊢ (⟨𝑦, 𝑧⟩ ∈ ∪ 𝐴 ↔ ∃𝑥(⟨𝑦, 𝑧⟩ ∈ 𝑥 ∧ 𝑥 ∈ 𝐴))
11	10	exbii 1849	. . . 4 ⊢ (∃𝑧⟨𝑦, 𝑧⟩ ∈ ∪ 𝐴 ↔ ∃𝑧∃𝑥(⟨𝑦, 𝑧⟩ ∈ 𝑥 ∧ 𝑥 ∈ 𝐴))
12		df-rex 3059	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ dom 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ dom 𝑥))
13	9, 11, 12	3bitr4i 303	. . 3 ⊢ (∃𝑧⟨𝑦, 𝑧⟩ ∈ ∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ dom 𝑥)
14	4	eldm2 5848	. . 3 ⊢ (𝑦 ∈ dom ∪ 𝐴 ↔ ∃𝑧⟨𝑦, 𝑧⟩ ∈ ∪ 𝐴)
15		eliun 4947	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 dom 𝑥 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ dom 𝑥)
16	13, 14, 15	3bitr4i 303	. 2 ⊢ (𝑦 ∈ dom ∪ 𝐴 ↔ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 dom 𝑥)
17	16	eqriv 2730	1 ⊢ dom ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 dom 𝑥

Syntax hints:   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2113  ∃wrex 3058  ⟨cop 4583  ∪ cuni 4860  ∪ ciun 4943  dom cdm 5621

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-11 2162  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-dm 5631

This theorem is referenced by:  frrlem7  8231  tfrlem8  8312  axdc3lem2  10352  bnj1400  34858