Theorem ordtuni 23199

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 7924	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → dom 𝑅 ∈ V)
3	1, 2	eqeltrid 2844	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → 𝑋 ∈ V)
4		unisng 4924	. . . . 5 ⊢ (𝑋 ∈ V → ∪ {𝑋} = 𝑋)
5	3, 4	syl 17	. . . 4 ⊢ (𝑅 ∈ 𝑉 → ∪ {𝑋} = 𝑋)
6	5	uneq1d 4166	. . 3 ⊢ (𝑅 ∈ 𝑉 → (∪ {𝑋} ∪ ∪ (𝐴 ∪ 𝐵)) = (𝑋 ∪ ∪ (𝐴 ∪ 𝐵)))
7		ordtval.2	. . . . . . 7 ⊢ 𝐴 = ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})
8		ssrab2 4079	. . . . . . . . . 10 ⊢ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} ⊆ 𝑋
9	3	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑥 ∈ 𝑋) → 𝑋 ∈ V)
10		elpw2g 5332	. . . . . . . . . . 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 7134	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}):𝑋⟶𝒫 𝑋)
14	13	frnd 6743	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) ⊆ 𝒫 𝑋)
15	7, 14	eqsstrid 4021	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → 𝐴 ⊆ 𝒫 𝑋)
16		ordtval.3	. . . . . . 7 ⊢ 𝐵 = ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦})
17		ssrab2 4079	. . . . . . . . . 10 ⊢ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦} ⊆ 𝑋
18		elpw2g 5332	. . . . . . . . . . 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 7134	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}):𝑋⟶𝒫 𝑋)
22	21	frnd 6743	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ⊆ 𝒫 𝑋)
23	16, 22	eqsstrid 4021	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → 𝐵 ⊆ 𝒫 𝑋)
24	15, 23	unssd 4191	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (𝐴 ∪ 𝐵) ⊆ 𝒫 𝑋)
25		sspwuni 5099	. . . . 5 ⊢ ((𝐴 ∪ 𝐵) ⊆ 𝒫 𝑋 ↔ ∪ (𝐴 ∪ 𝐵) ⊆ 𝑋)
26	24, 25	sylib 218	. . . 4 ⊢ (𝑅 ∈ 𝑉 → ∪ (𝐴 ∪ 𝐵) ⊆ 𝑋)
27		ssequn2 4188	. . . 4 ⊢ (∪ (𝐴 ∪ 𝐵) ⊆ 𝑋 ↔ (𝑋 ∪ ∪ (𝐴 ∪ 𝐵)) = 𝑋)
28	26, 27	sylib 218	. . 3 ⊢ (𝑅 ∈ 𝑉 → (𝑋 ∪ ∪ (𝐴 ∪ 𝐵)) = 𝑋)
29	6, 28	eqtr2d 2777	. 2 ⊢ (𝑅 ∈ 𝑉 → 𝑋 = (∪ {𝑋} ∪ ∪ (𝐴 ∪ 𝐵)))
30		uniun 4929	. 2 ⊢ ∪ ({𝑋} ∪ (𝐴 ∪ 𝐵)) = (∪ {𝑋} ∪ ∪ (𝐴 ∪ 𝐵))
31	29, 30	eqtr4di 2794	1 ⊢ (𝑅 ∈ 𝑉 → 𝑋 = ∪ ({𝑋} ∪ (𝐴 ∪ 𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539   ∈ wcel 2107  {crab 3435  Vcvv 3479   ∪ cun 3948   ⊆ wss 3950  𝒫 cpw 4599  {csn 4625  ∪ cuni 4906   class class class wbr 5142   ↦ cmpt 5224  dom cdm 5684  ran crn 5685

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-fun 6562  df-fn 6563  df-f 6564

This theorem is referenced by:  ordtbas2  23200  ordtbas  23201  ordttopon  23202  ordtopn1  23203  ordtopn2  23204  ordtrest2  23213  ordthmeolem  23810  ordtprsuni  33919