Theorem dmrnssfld 5962

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 3467	. . . . 5 ⊢ 𝑥 ∈ V
2	1	eldm2 5889	. . . 4 ⊢ (𝑥 ∈ dom 𝐴 ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴)
3	1	prid1 4730	. . . . . 6 ⊢ 𝑥 ∈ {𝑥, 𝑦}
4		vex 3467	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
5	1, 4	uniop 5496	. . . . . . . . 9 ⊢ ∪ ⟨𝑥, 𝑦⟩ = {𝑥, 𝑦}
6	1, 4	uniopel 5497	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → ∪ ⟨𝑥, 𝑦⟩ ∈ ∪ 𝐴)
7	5, 6	eqeltrrid 2874	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → {𝑥, 𝑦} ∈ ∪ 𝐴)
8		elssuni 4905	. . . . . . . 8 ⊢ ({𝑥, 𝑦} ∈ ∪ 𝐴 → {𝑥, 𝑦} ⊆ ∪ ∪ 𝐴)
9	7, 8	syl 18	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → {𝑥, 𝑦} ⊆ ∪ ∪ 𝐴)
10	9	sseld 3944	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → (𝑥 ∈ {𝑥, 𝑦} → 𝑥 ∈ ∪ ∪ 𝐴))
11	3, 10	mpi 21	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑥 ∈ ∪ ∪ 𝐴)
12	11	exlimiv 1957	. . . 4 ⊢ (∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑥 ∈ ∪ ∪ 𝐴)
13	2, 12	sylbi 220	. . 3 ⊢ (𝑥 ∈ dom 𝐴 → 𝑥 ∈ ∪ ∪ 𝐴)
14	13	ssriv 3949	. 2 ⊢ dom 𝐴 ⊆ ∪ ∪ 𝐴
15	4	elrn2 5880	. . . 4 ⊢ (𝑦 ∈ ran 𝐴 ↔ ∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐴)
16	4	prid2 4731	. . . . . 6 ⊢ 𝑦 ∈ {𝑥, 𝑦}
17	9	sseld 3944	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → (𝑦 ∈ {𝑥, 𝑦} → 𝑦 ∈ ∪ ∪ 𝐴))
18	16, 17	mpi 21	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑦 ∈ ∪ ∪ 𝐴)
19	18	exlimiv 1957	. . . 4 ⊢ (∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑦 ∈ ∪ ∪ 𝐴)
20	15, 19	sylbi 220	. . 3 ⊢ (𝑦 ∈ ran 𝐴 → 𝑦 ∈ ∪ ∪ 𝐴)
21	20	ssriv 3949	. 2 ⊢ ran 𝐴 ⊆ ∪ ∪ 𝐴
22	14, 21	unssi 4152	1 ⊢ (dom 𝐴 ∪ ran 𝐴) ⊆ ∪ ∪ 𝐴

Syntax hints:  ∃wex 1806   ∈ wcel 2149   ∪ cun 3911   ⊆ wss 3913  {cpr 4593  ⟨cop 4597  ∪ cuni 4873  dom cdm 5659  ran crn 5660

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5258  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-opab 5175  df-cnv 5667  df-dm 5669  df-rn 5670

This theorem is referenced by:  relfld  6273  relcoi2  6275  dmexg  7894  rnexg  7895  wundm  10709  wunrn  10710  relexpdm  15076  relexprn  15080  relexpfld  15082  psdmrn  18625  dirdm  18652  dirge  18655  tailf  36771  filnetlem3  36776  dmwf  45559  rnwf  45560