Theorem dmuni 5334

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 2042	. . . . 5 ⊢ (∃𝑧∃𝑥(⟨𝑦, 𝑧⟩ ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) ↔ ∃𝑥∃𝑧(⟨𝑦, 𝑧⟩ ∈ 𝑥 ∧ 𝑥 ∈ 𝐴))
2		ancom 466	. . . . . . 7 ⊢ ((∃𝑧⟨𝑦, 𝑧⟩ ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑧⟨𝑦, 𝑧⟩ ∈ 𝑥))
3		19.41v 1914	. . . . . . 7 ⊢ (∃𝑧(⟨𝑦, 𝑧⟩ ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) ↔ (∃𝑧⟨𝑦, 𝑧⟩ ∈ 𝑥 ∧ 𝑥 ∈ 𝐴))
4		vex 3203	. . . . . . . . 9 ⊢ 𝑦 ∈ V
5	4	eldm2 5322	. . . . . . . 8 ⊢ (𝑦 ∈ dom 𝑥 ↔ ∃𝑧⟨𝑦, 𝑧⟩ ∈ 𝑥)
6	5	anbi2i 730	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ dom 𝑥) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑧⟨𝑦, 𝑧⟩ ∈ 𝑥))
7	2, 3, 6	3bitr4i 292	. . . . . 6 ⊢ (∃𝑧(⟨𝑦, 𝑧⟩ ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ dom 𝑥))
8	7	exbii 1774	. . . . 5 ⊢ (∃𝑥∃𝑧(⟨𝑦, 𝑧⟩ ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ dom 𝑥))
9	1, 8	bitri 264	. . . 4 ⊢ (∃𝑧∃𝑥(⟨𝑦, 𝑧⟩ ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ dom 𝑥))
10		eluni 4439	. . . . 5 ⊢ (⟨𝑦, 𝑧⟩ ∈ ∪ 𝐴 ↔ ∃𝑥(⟨𝑦, 𝑧⟩ ∈ 𝑥 ∧ 𝑥 ∈ 𝐴))
11	10	exbii 1774	. . . 4 ⊢ (∃𝑧⟨𝑦, 𝑧⟩ ∈ ∪ 𝐴 ↔ ∃𝑧∃𝑥(⟨𝑦, 𝑧⟩ ∈ 𝑥 ∧ 𝑥 ∈ 𝐴))
12		df-rex 2918	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ dom 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ dom 𝑥))
13	9, 11, 12	3bitr4i 292	. . 3 ⊢ (∃𝑧⟨𝑦, 𝑧⟩ ∈ ∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ dom 𝑥)
14	4	eldm2 5322	. . 3 ⊢ (𝑦 ∈ dom ∪ 𝐴 ↔ ∃𝑧⟨𝑦, 𝑧⟩ ∈ ∪ 𝐴)
15		eliun 4524	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 dom 𝑥 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ dom 𝑥)
16	13, 14, 15	3bitr4i 292	. 2 ⊢ (𝑦 ∈ dom ∪ 𝐴 ↔ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 dom 𝑥)
17	16	eqriv 2619	1 ⊢ dom ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 dom 𝑥

Syntax hints:   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990  ∃wrex 2913  ⟨cop 4183  ∪ cuni 4436  ∪ ciun 4520  dom cdm 5114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-dm 5124

This theorem is referenced by:  wfrdmss  7421  wfrdmcl  7423  tfrlem8  7480  axdc3lem2  9273  bnj1400  30906  frrlem5d  31787  frrlem5e  31788  frrlem7  31790