Theorem ordtuni 23105

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 7831	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → dom 𝑅 ∈ V)
3	1, 2	eqeltrid 2835	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → 𝑋 ∈ V)
4		unisng 4874	. . . . 5 ⊢ (𝑋 ∈ V → ∪ {𝑋} = 𝑋)
5	3, 4	syl 17	. . . 4 ⊢ (𝑅 ∈ 𝑉 → ∪ {𝑋} = 𝑋)
6	5	uneq1d 4114	. . 3 ⊢ (𝑅 ∈ 𝑉 → (∪ {𝑋} ∪ ∪ (𝐴 ∪ 𝐵)) = (𝑋 ∪ ∪ (𝐴 ∪ 𝐵)))
7		ordtval.2	. . . . . . 7 ⊢ 𝐴 = ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})
8		ssrab2 4027	. . . . . . . . . 10 ⊢ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} ⊆ 𝑋
9	3	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑥 ∈ 𝑋) → 𝑋 ∈ V)
10		elpw2g 5269	. . . . . . . . . . 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 7048	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}):𝑋⟶𝒫 𝑋)
14	13	frnd 6659	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) ⊆ 𝒫 𝑋)
15	7, 14	eqsstrid 3968	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → 𝐴 ⊆ 𝒫 𝑋)
16		ordtval.3	. . . . . . 7 ⊢ 𝐵 = ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦})
17		ssrab2 4027	. . . . . . . . . 10 ⊢ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦} ⊆ 𝑋
18		elpw2g 5269	. . . . . . . . . . 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 7048	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}):𝑋⟶𝒫 𝑋)
22	21	frnd 6659	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ⊆ 𝒫 𝑋)
23	16, 22	eqsstrid 3968	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → 𝐵 ⊆ 𝒫 𝑋)
24	15, 23	unssd 4139	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (𝐴 ∪ 𝐵) ⊆ 𝒫 𝑋)
25		sspwuni 5046	. . . . 5 ⊢ ((𝐴 ∪ 𝐵) ⊆ 𝒫 𝑋 ↔ ∪ (𝐴 ∪ 𝐵) ⊆ 𝑋)
26	24, 25	sylib 218	. . . 4 ⊢ (𝑅 ∈ 𝑉 → ∪ (𝐴 ∪ 𝐵) ⊆ 𝑋)
27		ssequn2 4136	. . . 4 ⊢ (∪ (𝐴 ∪ 𝐵) ⊆ 𝑋 ↔ (𝑋 ∪ ∪ (𝐴 ∪ 𝐵)) = 𝑋)
28	26, 27	sylib 218	. . 3 ⊢ (𝑅 ∈ 𝑉 → (𝑋 ∪ ∪ (𝐴 ∪ 𝐵)) = 𝑋)
29	6, 28	eqtr2d 2767	. 2 ⊢ (𝑅 ∈ 𝑉 → 𝑋 = (∪ {𝑋} ∪ ∪ (𝐴 ∪ 𝐵)))
30		uniun 4879	. 2 ⊢ ∪ ({𝑋} ∪ (𝐴 ∪ 𝐵)) = (∪ {𝑋} ∪ ∪ (𝐴 ∪ 𝐵))
31	29, 30	eqtr4di 2784	1 ⊢ (𝑅 ∈ 𝑉 → 𝑋 = ∪ ({𝑋} ∪ (𝐴 ∪ 𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  {crab 3395  Vcvv 3436   ∪ cun 3895   ⊆ wss 3897  𝒫 cpw 4547  {csn 4573  ∪ cuni 4856   class class class wbr 5089   ↦ cmpt 5170  dom cdm 5614  ran crn 5615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-fun 6483  df-fn 6484  df-f 6485

This theorem is referenced by:  ordtbas2  23106  ordtbas  23107  ordttopon  23108  ordtopn1  23109  ordtopn2  23110  ordtrest2  23119  ordthmeolem  23716  ordtprsuni  33932