Theorem dmrnssfld 5305

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 3176	. . . . 5 ⊢ 𝑥 ∈ V
2	1	eldm2 5244	. . . 4 ⊢ (𝑥 ∈ dom 𝐴 ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴)
3	1	prid1 4241	. . . . . 6 ⊢ 𝑥 ∈ {𝑥, 𝑦}
4		vex 3176	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
5	1, 4	uniop 4902	. . . . . . . . 9 ⊢ ∪ ⟨𝑥, 𝑦⟩ = {𝑥, 𝑦}
6	1, 4	uniopel 4903	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ∪ ⟨𝑥, 𝑦⟩ ∈ ∪ 𝐴)
7	5, 6	syl5eqelr 2693	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → {𝑥, 𝑦} ∈ ∪ 𝐴)
8		elssuni 4403	. . . . . . . 8 ⊢ ({𝑥, 𝑦} ∈ ∪ 𝐴 → {𝑥, 𝑦} ⊆ ∪ ∪ 𝐴)
9	7, 8	syl 17	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → {𝑥, 𝑦} ⊆ ∪ ∪ 𝐴)
10	9	sseld 3567	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → (𝑥 ∈ {𝑥, 𝑦} → 𝑥 ∈ ∪ ∪ 𝐴))
11	3, 10	mpi 20	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑥 ∈ ∪ ∪ 𝐴)
12	11	exlimiv 1845	. . . 4 ⊢ (∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑥 ∈ ∪ ∪ 𝐴)
13	2, 12	sylbi 206	. . 3 ⊢ (𝑥 ∈ dom 𝐴 → 𝑥 ∈ ∪ ∪ 𝐴)
14	13	ssriv 3572	. 2 ⊢ dom 𝐴 ⊆ ∪ ∪ 𝐴
15	4	elrn2 5286	. . . 4 ⊢ (𝑦 ∈ ran 𝐴 ↔ ∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐴)
16	4	prid2 4242	. . . . . 6 ⊢ 𝑦 ∈ {𝑥, 𝑦}
17	9	sseld 3567	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → (𝑦 ∈ {𝑥, 𝑦} → 𝑦 ∈ ∪ ∪ 𝐴))
18	16, 17	mpi 20	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑦 ∈ ∪ ∪ 𝐴)
19	18	exlimiv 1845	. . . 4 ⊢ (∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑦 ∈ ∪ ∪ 𝐴)
20	15, 19	sylbi 206	. . 3 ⊢ (𝑦 ∈ ran 𝐴 → 𝑦 ∈ ∪ ∪ 𝐴)
21	20	ssriv 3572	. 2 ⊢ ran 𝐴 ⊆ ∪ ∪ 𝐴
22	14, 21	unssi 3750	1 ⊢ (dom 𝐴 ∪ ran 𝐴) ⊆ ∪ ∪ 𝐴

Syntax hints:  ∃wex 1695   ∈ wcel 1977   ∪ cun 3538   ⊆ wss 3540  {cpr 4127  ⟨cop 4131  ∪ cuni 4372  dom cdm 5038  ran crn 5039

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-cnv 5046  df-dm 5048  df-rn 5049

This theorem is referenced by:  relfld  5578  relcoi2  5580  dmexg  6989  rnexg  6990  wundm  9429  wunrn  9430  relexpdm  13631  relexprn  13635  relexpfld  13637  psdmrn  17030  dirdm  17057  dirge  17060  tailf  31540  filnetlem3  31545