Theorem dmuni 5905

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 2154	. . . . 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 1945	. . . . . . 7 ⊢ (∃𝑧(⟨𝑦, 𝑧⟩ ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) ↔ (∃𝑧⟨𝑦, 𝑧⟩ ∈ 𝑥 ∧ 𝑥 ∈ 𝐴))
4		vex 3470	. . . . . . . . 9 ⊢ 𝑦 ∈ V
5	4	eldm2 5892	. . . . . . . 8 ⊢ (𝑦 ∈ dom 𝑥 ↔ ∃𝑧⟨𝑦, 𝑧⟩ ∈ 𝑥)
6	5	anbi2i 622	. . . . . . 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 1842	. . . . 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 4903	. . . . 5 ⊢ (⟨𝑦, 𝑧⟩ ∈ ∪ 𝐴 ↔ ∃𝑥(⟨𝑦, 𝑧⟩ ∈ 𝑥 ∧ 𝑥 ∈ 𝐴))
11	10	exbii 1842	. . . 4 ⊢ (∃𝑧⟨𝑦, 𝑧⟩ ∈ ∪ 𝐴 ↔ ∃𝑧∃𝑥(⟨𝑦, 𝑧⟩ ∈ 𝑥 ∧ 𝑥 ∈ 𝐴))
12		df-rex 3063	. . . 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 5892	. . 3 ⊢ (𝑦 ∈ dom ∪ 𝐴 ↔ ∃𝑧⟨𝑦, 𝑧⟩ ∈ ∪ 𝐴)
15		eliun 4992	. . 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 2721	1 ⊢ dom ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 dom 𝑥

Syntax hints:   ∧ wa 395   = wceq 1533  ∃wex 1773   ∈ wcel 2098  ∃wrex 3062  ⟨cop 4627  ∪ cuni 4900  ∪ ciun 4988  dom cdm 5667

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-11 2146  ax-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-dm 5677

This theorem is referenced by:  frrlem7  8273  wfrdmssOLD  8311  wfrdmclOLD  8313  tfrlem8  8380  axdc3lem2  10443  bnj1400  34364