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 3454	. . . . 5 ⊢ 𝑥 ∈ V
2	1	eldm2 5867	. . . 4 ⊢ (𝑥 ∈ dom 𝐴 ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴)
3	1	prid1 4728	. . . . . 6 ⊢ 𝑥 ∈ {𝑥, 𝑦}
4		vex 3454	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
5	1, 4	uniop 5477	. . . . . . . . 9 ⊢ ∪ ⟨𝑥, 𝑦⟩ = {𝑥, 𝑦}
6	1, 4	uniopel 5478	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ∪ ⟨𝑥, 𝑦⟩ ∈ ∪ 𝐴)
7	5, 6	eqeltrrid 2834	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → {𝑥, 𝑦} ∈ ∪ 𝐴)
8		elssuni 4903	. . . . . . . 8 ⊢ ({𝑥, 𝑦} ∈ ∪ 𝐴 → {𝑥, 𝑦} ⊆ ∪ ∪ 𝐴)
9	7, 8	syl 17	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → {𝑥, 𝑦} ⊆ ∪ ∪ 𝐴)
10	9	sseld 3947	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → (𝑥 ∈ {𝑥, 𝑦} → 𝑥 ∈ ∪ ∪ 𝐴))
11	3, 10	mpi 20	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑥 ∈ ∪ ∪ 𝐴)
12	11	exlimiv 1930	. . . 4 ⊢ (∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑥 ∈ ∪ ∪ 𝐴)
13	2, 12	sylbi 217	. . 3 ⊢ (𝑥 ∈ dom 𝐴 → 𝑥 ∈ ∪ ∪ 𝐴)
14	13	ssriv 3952	. 2 ⊢ dom 𝐴 ⊆ ∪ ∪ 𝐴
15	4	elrn2 5858	. . . 4 ⊢ (𝑦 ∈ ran 𝐴 ↔ ∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐴)
16	4	prid2 4729	. . . . . 6 ⊢ 𝑦 ∈ {𝑥, 𝑦}
17	9	sseld 3947	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → (𝑦 ∈ {𝑥, 𝑦} → 𝑦 ∈ ∪ ∪ 𝐴))
18	16, 17	mpi 20	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑦 ∈ ∪ ∪ 𝐴)
19	18	exlimiv 1930	. . . 4 ⊢ (∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑦 ∈ ∪ ∪ 𝐴)
20	15, 19	sylbi 217	. . 3 ⊢ (𝑦 ∈ ran 𝐴 → 𝑦 ∈ ∪ ∪ 𝐴)
21	20	ssriv 3952	. 2 ⊢ ran 𝐴 ⊆ ∪ ∪ 𝐴
22	14, 21	unssi 4156	1 ⊢ (dom 𝐴 ∪ ran 𝐴) ⊆ ∪ ∪ 𝐴

Syntax hints:  ∃wex 1779   ∈ wcel 2109   ∪ cun 3914   ⊆ wss 3916  {cpr 4593  ⟨cop 4597  ∪ cuni 4873  dom cdm 5640  ran crn 5641

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5253  ax-nul 5263  ax-pr 5389

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-rab 3409  df-v 3452  df-dif 3919  df-un 3921  df-ss 3933  df-nul 4299  df-if 4491  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5110  df-opab 5172  df-cnv 5648  df-dm 5650  df-rn 5651

This theorem is referenced by:  relfld  6250  relcoi2  6252  dmexg  7879  rnexg  7880  wundm  10687  wunrn  10688  relexpdm  15015  relexprn  15019  relexpfld  15021  psdmrn  18538  dirdm  18565  dirge  18568  tailf  36358  filnetlem3  36363  dmwf  44948  rnwf  44949