Theorem dmrnssfld 5939

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 3448	. . . . 5 ⊢ 𝑥 ∈ V
2	1	eldm2 5866	. . . 4 ⊢ (𝑥 ∈ dom 𝐴 ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴)
3	1	prid1 4711	. . . . . 6 ⊢ 𝑥 ∈ {𝑥, 𝑦}
4		vex 3448	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
5	1, 4	uniop 5474	. . . . . . . . 9 ⊢ ∪ ⟨𝑥, 𝑦⟩ = {𝑥, 𝑦}
6	1, 4	uniopel 5475	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ∪ ⟨𝑥, 𝑦⟩ ∈ ∪ 𝐴)
7	5, 6	eqeltrrid 2857	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → {𝑥, 𝑦} ∈ ∪ 𝐴)
8		elssuni 4887	. . . . . . . 8 ⊢ ({𝑥, 𝑦} ∈ ∪ 𝐴 → {𝑥, 𝑦} ⊆ ∪ ∪ 𝐴)
9	7, 8	syl 17	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → {𝑥, 𝑦} ⊆ ∪ ∪ 𝐴)
10	9	sseld 3926	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → (𝑥 ∈ {𝑥, 𝑦} → 𝑥 ∈ ∪ ∪ 𝐴))
11	3, 10	mpi 20	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑥 ∈ ∪ ∪ 𝐴)
12	11	exlimiv 1940	. . . 4 ⊢ (∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑥 ∈ ∪ ∪ 𝐴)
13	2, 12	sylbi 219	. . 3 ⊢ (𝑥 ∈ dom 𝐴 → 𝑥 ∈ ∪ ∪ 𝐴)
14	13	ssriv 3931	. 2 ⊢ dom 𝐴 ⊆ ∪ ∪ 𝐴
15	4	elrn2 5857	. . . 4 ⊢ (𝑦 ∈ ran 𝐴 ↔ ∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐴)
16	4	prid2 4712	. . . . . 6 ⊢ 𝑦 ∈ {𝑥, 𝑦}
17	9	sseld 3926	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → (𝑦 ∈ {𝑥, 𝑦} → 𝑦 ∈ ∪ ∪ 𝐴))
18	16, 17	mpi 20	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑦 ∈ ∪ ∪ 𝐴)
19	18	exlimiv 1940	. . . 4 ⊢ (∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑦 ∈ ∪ ∪ 𝐴)
20	15, 19	sylbi 219	. . 3 ⊢ (𝑦 ∈ ran 𝐴 → 𝑦 ∈ ∪ ∪ 𝐴)
21	20	ssriv 3931	. 2 ⊢ ran 𝐴 ⊆ ∪ ∪ 𝐴
22	14, 21	unssi 4134	1 ⊢ (dom 𝐴 ∪ ran 𝐴) ⊆ ∪ ∪ 𝐴

Syntax hints:  ∃wex 1789   ∈ wcel 2132   ∪ cun 3893   ⊆ wss 3895  {cpr 4574  ⟨cop 4578  ∪ cuni 4855  dom cdm 5636  ran crn 5637

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-ext 2724  ax-sep 5236  ax-pr 5380

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-sb 2081  df-clab 2731  df-cleq 2744  df-clel 2827  df-rab 3405  df-v 3446  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-nul 4277  df-if 4471  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4856  df-br 5091  df-opab 5153  df-cnv 5644  df-dm 5646  df-rn 5647

This theorem is referenced by:  relfld  6247  relcoi2  6249  dmexg  7867  rnexg  7868  wundm  10672  wunrn  10673  relexpdm  15042  relexprn  15046  relexpfld  15048  psdmrn  18577  dirdm  18604  dirge  18607  tailf  36673  filnetlem3  36678  dmwf  45479  rnwf  45480