Theorem ordtuni 23133

Description: Value of the order topology. (Contributed by Mario Carneiro, 3-Sep-2015.)

Hypotheses

Ref	Expression
ordtval.1	⊢ 𝑋 = dom 𝑅
ordtval.2	⊢ 𝐴 = ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})
ordtval.3	⊢ 𝐵 = ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦})

Assertion

Ref	Expression
ordtuni	⊢ (𝑅 ∈ 𝑉 → 𝑋 = ∪ ({𝑋} ∪ (𝐴 ∪ 𝐵)))

Distinct variable groups:   𝑥,𝑦,𝑅   𝑥,𝑋,𝑦   𝑥,𝑉

Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝑉(𝑦)

Proof of Theorem ordtuni

Step	Hyp	Ref	Expression
1		ordtval.1	. . . . . 6 ⊢ 𝑋 = dom 𝑅
2		dmexg 7902	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → dom 𝑅 ∈ V)
3	1, 2	eqeltrid 2839	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → 𝑋 ∈ V)
4		unisng 4906	. . . . 5 ⊢ (𝑋 ∈ V → ∪ {𝑋} = 𝑋)
5	3, 4	syl 17	. . . 4 ⊢ (𝑅 ∈ 𝑉 → ∪ {𝑋} = 𝑋)
6	5	uneq1d 4147	. . 3 ⊢ (𝑅 ∈ 𝑉 → (∪ {𝑋} ∪ ∪ (𝐴 ∪ 𝐵)) = (𝑋 ∪ ∪ (𝐴 ∪ 𝐵)))
7		ordtval.2	. . . . . . 7 ⊢ 𝐴 = ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})
8		ssrab2 4060	. . . . . . . . . 10 ⊢ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} ⊆ 𝑋
9	3	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑥 ∈ 𝑋) → 𝑋 ∈ V)
10		elpw2g 5308	. . . . . . . . . . 11 ⊢ (𝑋 ∈ V → ({𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} ∈ 𝒫 𝑋 ↔ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} ⊆ 𝑋))
11	9, 10	syl 17	. . . . . . . . . 10 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑥 ∈ 𝑋) → ({𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} ∈ 𝒫 𝑋 ↔ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} ⊆ 𝑋))
12	8, 11	mpbiri 258	. . . . . . . . 9 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑥 ∈ 𝑋) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} ∈ 𝒫 𝑋)
13	12	fmpttd 7110	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}):𝑋⟶𝒫 𝑋)
14	13	frnd 6719	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) ⊆ 𝒫 𝑋)
15	7, 14	eqsstrid 4002	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → 𝐴 ⊆ 𝒫 𝑋)
16		ordtval.3	. . . . . . 7 ⊢ 𝐵 = ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦})
17		ssrab2 4060	. . . . . . . . . 10 ⊢ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦} ⊆ 𝑋
18		elpw2g 5308	. . . . . . . . . . 11 ⊢ (𝑋 ∈ V → ({𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦} ∈ 𝒫 𝑋 ↔ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦} ⊆ 𝑋))
19	9, 18	syl 17	. . . . . . . . . 10 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑥 ∈ 𝑋) → ({𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦} ∈ 𝒫 𝑋 ↔ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦} ⊆ 𝑋))
20	17, 19	mpbiri 258	. . . . . . . . 9 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑥 ∈ 𝑋) → {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦} ∈ 𝒫 𝑋)
21	20	fmpttd 7110	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}):𝑋⟶𝒫 𝑋)
22	21	frnd 6719	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ⊆ 𝒫 𝑋)
23	16, 22	eqsstrid 4002	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → 𝐵 ⊆ 𝒫 𝑋)
24	15, 23	unssd 4172	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (𝐴 ∪ 𝐵) ⊆ 𝒫 𝑋)
25		sspwuni 5081	. . . . 5 ⊢ ((𝐴 ∪ 𝐵) ⊆ 𝒫 𝑋 ↔ ∪ (𝐴 ∪ 𝐵) ⊆ 𝑋)
26	24, 25	sylib 218	. . . 4 ⊢ (𝑅 ∈ 𝑉 → ∪ (𝐴 ∪ 𝐵) ⊆ 𝑋)
27		ssequn2 4169	. . . 4 ⊢ (∪ (𝐴 ∪ 𝐵) ⊆ 𝑋 ↔ (𝑋 ∪ ∪ (𝐴 ∪ 𝐵)) = 𝑋)
28	26, 27	sylib 218	. . 3 ⊢ (𝑅 ∈ 𝑉 → (𝑋 ∪ ∪ (𝐴 ∪ 𝐵)) = 𝑋)
29	6, 28	eqtr2d 2772	. 2 ⊢ (𝑅 ∈ 𝑉 → 𝑋 = (∪ {𝑋} ∪ ∪ (𝐴 ∪ 𝐵)))
30		uniun 4911	. 2 ⊢ ∪ ({𝑋} ∪ (𝐴 ∪ 𝐵)) = (∪ {𝑋} ∪ ∪ (𝐴 ∪ 𝐵))
31	29, 30	eqtr4di 2789	1 ⊢ (𝑅 ∈ 𝑉 → 𝑋 = ∪ ({𝑋} ∪ (𝐴 ∪ 𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {crab 3420  Vcvv 3464   ∪ cun 3929   ⊆ wss 3931  𝒫 cpw 4580  {csn 4606  ∪ cuni 4888   class class class wbr 5124   ↦ cmpt 5206  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 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407  ax-un 7734

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-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-fun 6538  df-fn 6539  df-f 6540

This theorem is referenced by:  ordtbas2  23134  ordtbas  23135  ordttopon  23136  ordtopn1  23137  ordtopn2  23138  ordtrest2  23147  ordthmeolem  23744  ordtprsuni  33955