Theorem ordtuni 23132

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 7841	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → dom 𝑅 ∈ V)
3	1, 2	eqeltrid 2838	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → 𝑋 ∈ V)
4		unisng 4879	. . . . 5 ⊢ (𝑋 ∈ V → ∪ {𝑋} = 𝑋)
5	3, 4	syl 17	. . . 4 ⊢ (𝑅 ∈ 𝑉 → ∪ {𝑋} = 𝑋)
6	5	uneq1d 4117	. . 3 ⊢ (𝑅 ∈ 𝑉 → (∪ {𝑋} ∪ ∪ (𝐴 ∪ 𝐵)) = (𝑋 ∪ ∪ (𝐴 ∪ 𝐵)))
7		ordtval.2	. . . . . . 7 ⊢ 𝐴 = ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥})
8		ssrab2 4030	. . . . . . . . . 10 ⊢ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥} ⊆ 𝑋
9	3	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑥 ∈ 𝑋) → 𝑋 ∈ V)
10		elpw2g 5276	. . . . . . . . . . 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 7058	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}):𝑋⟶𝒫 𝑋)
14	13	frnd 6668	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑦𝑅𝑥}) ⊆ 𝒫 𝑋)
15	7, 14	eqsstrid 3970	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → 𝐴 ⊆ 𝒫 𝑋)
16		ordtval.3	. . . . . . 7 ⊢ 𝐵 = ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦})
17		ssrab2 4030	. . . . . . . . . 10 ⊢ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦} ⊆ 𝑋
18		elpw2g 5276	. . . . . . . . . . 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 7058	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}):𝑋⟶𝒫 𝑋)
22	21	frnd 6668	. . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → ran (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝑋 ∣ ¬ 𝑥𝑅𝑦}) ⊆ 𝒫 𝑋)
23	16, 22	eqsstrid 3970	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → 𝐵 ⊆ 𝒫 𝑋)
24	15, 23	unssd 4142	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (𝐴 ∪ 𝐵) ⊆ 𝒫 𝑋)
25		sspwuni 5053	. . . . 5 ⊢ ((𝐴 ∪ 𝐵) ⊆ 𝒫 𝑋 ↔ ∪ (𝐴 ∪ 𝐵) ⊆ 𝑋)
26	24, 25	sylib 218	. . . 4 ⊢ (𝑅 ∈ 𝑉 → ∪ (𝐴 ∪ 𝐵) ⊆ 𝑋)
27		ssequn2 4139	. . . 4 ⊢ (∪ (𝐴 ∪ 𝐵) ⊆ 𝑋 ↔ (𝑋 ∪ ∪ (𝐴 ∪ 𝐵)) = 𝑋)
28	26, 27	sylib 218	. . 3 ⊢ (𝑅 ∈ 𝑉 → (𝑋 ∪ ∪ (𝐴 ∪ 𝐵)) = 𝑋)
29	6, 28	eqtr2d 2770	. 2 ⊢ (𝑅 ∈ 𝑉 → 𝑋 = (∪ {𝑋} ∪ ∪ (𝐴 ∪ 𝐵)))
30		uniun 4884	. 2 ⊢ ∪ ({𝑋} ∪ (𝐴 ∪ 𝐵)) = (∪ {𝑋} ∪ ∪ (𝐴 ∪ 𝐵))
31	29, 30	eqtr4di 2787	1 ⊢ (𝑅 ∈ 𝑉 → 𝑋 = ∪ ({𝑋} ∪ (𝐴 ∪ 𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  {crab 3397  Vcvv 3438   ∪ cun 3897   ⊆ wss 3899  𝒫 cpw 4552  {csn 4578  ∪ cuni 4861   class class class wbr 5096   ↦ cmpt 5177  dom cdm 5622  ran crn 5623

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375  ax-un 7678

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 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-fun 6492  df-fn 6493  df-f 6494

This theorem is referenced by:  ordtbas2  23133  ordtbas  23134  ordttopon  23135  ordtopn1  23136  ordtopn2  23137  ordtrest2  23146  ordthmeolem  23743  ordtprsuni  34025