Theorem dmuni 5927

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 2159	. . . . 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 1946	. . . . . . 7 ⊢ (∃𝑧(⟨𝑦, 𝑧⟩ ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) ↔ (∃𝑧⟨𝑦, 𝑧⟩ ∈ 𝑥 ∧ 𝑥 ∈ 𝐴))
4		vex 3481	. . . . . . . . 9 ⊢ 𝑦 ∈ V
5	4	eldm2 5914	. . . . . . . 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 1844	. . . . 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 4914	. . . . 5 ⊢ (⟨𝑦, 𝑧⟩ ∈ ∪ 𝐴 ↔ ∃𝑥(⟨𝑦, 𝑧⟩ ∈ 𝑥 ∧ 𝑥 ∈ 𝐴))
11	10	exbii 1844	. . . 4 ⊢ (∃𝑧⟨𝑦, 𝑧⟩ ∈ ∪ 𝐴 ↔ ∃𝑧∃𝑥(⟨𝑦, 𝑧⟩ ∈ 𝑥 ∧ 𝑥 ∈ 𝐴))
12		df-rex 3068	. . . 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 5914	. . 3 ⊢ (𝑦 ∈ dom ∪ 𝐴 ↔ ∃𝑧⟨𝑦, 𝑧⟩ ∈ ∪ 𝐴)
15		eliun 4999	. . 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 2731	1 ⊢ dom ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 dom 𝑥

Syntax hints:   ∧ wa 395   = wceq 1536  ∃wex 1775   ∈ wcel 2105  ∃wrex 3067  ⟨cop 4636  ∪ cuni 4911  ∪ ciun 4995  dom cdm 5688

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-11 2154  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-dm 5698

This theorem is referenced by:  frrlem7  8315  wfrdmssOLD  8353  wfrdmclOLD  8355  tfrlem8  8422  axdc3lem2  10488  bnj1400  34827