Theorem dmuni 5661

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 2131	. . . . 5 ⊢ (∃𝑧∃𝑥(⟨𝑦, 𝑧⟩ ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) ↔ ∃𝑥∃𝑧(⟨𝑦, 𝑧⟩ ∈ 𝑥 ∧ 𝑥 ∈ 𝐴))
2		ancom 461	. . . . . . 7 ⊢ ((∃𝑧⟨𝑦, 𝑧⟩ ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑧⟨𝑦, 𝑧⟩ ∈ 𝑥))
3		19.41v 1925	. . . . . . 7 ⊢ (∃𝑧(⟨𝑦, 𝑧⟩ ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) ↔ (∃𝑧⟨𝑦, 𝑧⟩ ∈ 𝑥 ∧ 𝑥 ∈ 𝐴))
4		vex 3435	. . . . . . . . 9 ⊢ 𝑦 ∈ V
5	4	eldm2 5648	. . . . . . . 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 304	. . . . . 6 ⊢ (∃𝑧(⟨𝑦, 𝑧⟩ ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ dom 𝑥))
8	7	exbii 1827	. . . . 5 ⊢ (∃𝑥∃𝑧(⟨𝑦, 𝑧⟩ ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ dom 𝑥))
9	1, 8	bitri 276	. . . 4 ⊢ (∃𝑧∃𝑥(⟨𝑦, 𝑧⟩ ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ dom 𝑥))
10		eluni 4742	. . . . 5 ⊢ (⟨𝑦, 𝑧⟩ ∈ ∪ 𝐴 ↔ ∃𝑥(⟨𝑦, 𝑧⟩ ∈ 𝑥 ∧ 𝑥 ∈ 𝐴))
11	10	exbii 1827	. . . 4 ⊢ (∃𝑧⟨𝑦, 𝑧⟩ ∈ ∪ 𝐴 ↔ ∃𝑧∃𝑥(⟨𝑦, 𝑧⟩ ∈ 𝑥 ∧ 𝑥 ∈ 𝐴))
12		df-rex 3109	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ dom 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ dom 𝑥))
13	9, 11, 12	3bitr4i 304	. . 3 ⊢ (∃𝑧⟨𝑦, 𝑧⟩ ∈ ∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ dom 𝑥)
14	4	eldm2 5648	. . 3 ⊢ (𝑦 ∈ dom ∪ 𝐴 ↔ ∃𝑧⟨𝑦, 𝑧⟩ ∈ ∪ 𝐴)
15		eliun 4823	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 dom 𝑥 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ dom 𝑥)
16	13, 14, 15	3bitr4i 304	. 2 ⊢ (𝑦 ∈ dom ∪ 𝐴 ↔ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 dom 𝑥)
17	16	eqriv 2790	1 ⊢ dom ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 dom 𝑥

Syntax hints:   ∧ wa 396   = wceq 1520  ∃wex 1759   ∈ wcel 2079  ∃wrex 3104  ⟨cop 4472  ∪ cuni 4739  ∪ ciun 4819  dom cdm 5435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-ext 2767

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ral 3108  df-rex 3109  df-rab 3112  df-v 3434  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-nul 4207  df-if 4376  df-sn 4467  df-pr 4469  df-op 4473  df-uni 4740  df-iun 4821  df-br 4957  df-dm 5445

This theorem is referenced by:  wfrdmss  7804  wfrdmcl  7806  tfrlem8  7863  axdc3lem2  9708  bnj1400  31680  frrlem7  32683