Theorem dmrnssfld 4892

Description: The domain and range of a class are included in its double union. (Contributed by NM, 13-May-2008.)

Assertion

Ref	Expression
dmrnssfld	⊢ (dom 𝐴 ∪ ran 𝐴) ⊆ ∪ ∪ 𝐴

Proof of Theorem dmrnssfld

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

Step	Hyp	Ref	Expression
1		vex 2742	. . . . 5 ⊢ 𝑥 ∈ V
2	1	eldm2 4827	. . . 4 ⊢ (𝑥 ∈ dom 𝐴 ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴)
3	1	prid1 3700	. . . . . 6 ⊢ 𝑥 ∈ {𝑥, 𝑦}
4		vex 2742	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
5	1, 4	uniop 4257	. . . . . . . . 9 ⊢ ∪ ⟨𝑥, 𝑦⟩ = {𝑥, 𝑦}
6	1, 4	uniopel 4258	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ∪ ⟨𝑥, 𝑦⟩ ∈ ∪ 𝐴)
7	5, 6	eqeltrrid 2265	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → {𝑥, 𝑦} ∈ ∪ 𝐴)
8		elssuni 3839	. . . . . . . 8 ⊢ ({𝑥, 𝑦} ∈ ∪ 𝐴 → {𝑥, 𝑦} ⊆ ∪ ∪ 𝐴)
9	7, 8	syl 14	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → {𝑥, 𝑦} ⊆ ∪ ∪ 𝐴)
10	9	sseld 3156	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → (𝑥 ∈ {𝑥, 𝑦} → 𝑥 ∈ ∪ ∪ 𝐴))
11	3, 10	mpi 15	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑥 ∈ ∪ ∪ 𝐴)
12	11	exlimiv 1598	. . . 4 ⊢ (∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑥 ∈ ∪ ∪ 𝐴)
13	2, 12	sylbi 121	. . 3 ⊢ (𝑥 ∈ dom 𝐴 → 𝑥 ∈ ∪ ∪ 𝐴)
14	13	ssriv 3161	. 2 ⊢ dom 𝐴 ⊆ ∪ ∪ 𝐴
15	4	elrn2 4871	. . . 4 ⊢ (𝑦 ∈ ran 𝐴 ↔ ∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐴)
16	4	prid2 3701	. . . . . 6 ⊢ 𝑦 ∈ {𝑥, 𝑦}
17	9	sseld 3156	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → (𝑦 ∈ {𝑥, 𝑦} → 𝑦 ∈ ∪ ∪ 𝐴))
18	16, 17	mpi 15	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑦 ∈ ∪ ∪ 𝐴)
19	18	exlimiv 1598	. . . 4 ⊢ (∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑦 ∈ ∪ ∪ 𝐴)
20	15, 19	sylbi 121	. . 3 ⊢ (𝑦 ∈ ran 𝐴 → 𝑦 ∈ ∪ ∪ 𝐴)
21	20	ssriv 3161	. 2 ⊢ ran 𝐴 ⊆ ∪ ∪ 𝐴
22	14, 21	unssi 3312	1 ⊢ (dom 𝐴 ∪ ran 𝐴) ⊆ ∪ ∪ 𝐴

Syntax hints:  ∃wex 1492   ∈ wcel 2148   ∪ cun 3129   ⊆ wss 3131  {cpr 3595  ⟨cop 3597  ∪ cuni 3811  dom cdm 4628  ran crn 4629

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-cnv 4636  df-dm 4638  df-rn 4639

This theorem is referenced by:  dmexg  4893  rnexg  4894  relfld  5159  relcoi2  5161