Theorem topmeet 36382

Description: Two equivalent formulations of the meet of a collection of topologies. (Contributed by Jeff Hankins, 4-Oct-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)

Assertion

Ref	Expression
topmeet	⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → (𝒫 𝑋 ∩ ∩ 𝑆) = ∪ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗 ∈ 𝑆 𝑘 ⊆ 𝑗})

Distinct variable groups:   𝑗,𝑘,𝑆   𝑗,𝑉,𝑘   𝑗,𝑋,𝑘

Proof of Theorem topmeet

Step	Hyp	Ref	Expression
1		topmtcl 36381	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → (𝒫 𝑋 ∩ ∩ 𝑆) ∈ (TopOn‘𝑋))
2		inss2 4213	. . . . . . 7 ⊢ (𝒫 𝑋 ∩ ∩ 𝑆) ⊆ ∩ 𝑆
3		intss1 4939	. . . . . . 7 ⊢ (𝑗 ∈ 𝑆 → ∩ 𝑆 ⊆ 𝑗)
4	2, 3	sstrid 3970	. . . . . 6 ⊢ (𝑗 ∈ 𝑆 → (𝒫 𝑋 ∩ ∩ 𝑆) ⊆ 𝑗)
5	4	rgen 3053	. . . . 5 ⊢ ∀𝑗 ∈ 𝑆 (𝒫 𝑋 ∩ ∩ 𝑆) ⊆ 𝑗
6		sseq1 3984	. . . . . . 7 ⊢ (𝑘 = (𝒫 𝑋 ∩ ∩ 𝑆) → (𝑘 ⊆ 𝑗 ↔ (𝒫 𝑋 ∩ ∩ 𝑆) ⊆ 𝑗))
7	6	ralbidv 3163	. . . . . 6 ⊢ (𝑘 = (𝒫 𝑋 ∩ ∩ 𝑆) → (∀𝑗 ∈ 𝑆 𝑘 ⊆ 𝑗 ↔ ∀𝑗 ∈ 𝑆 (𝒫 𝑋 ∩ ∩ 𝑆) ⊆ 𝑗))
8	7	elrab 3671	. . . . 5 ⊢ ((𝒫 𝑋 ∩ ∩ 𝑆) ∈ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗 ∈ 𝑆 𝑘 ⊆ 𝑗} ↔ ((𝒫 𝑋 ∩ ∩ 𝑆) ∈ (TopOn‘𝑋) ∧ ∀𝑗 ∈ 𝑆 (𝒫 𝑋 ∩ ∩ 𝑆) ⊆ 𝑗))
9	5, 8	mpbiran2 710	. . . 4 ⊢ ((𝒫 𝑋 ∩ ∩ 𝑆) ∈ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗 ∈ 𝑆 𝑘 ⊆ 𝑗} ↔ (𝒫 𝑋 ∩ ∩ 𝑆) ∈ (TopOn‘𝑋))
10	1, 9	sylibr 234	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → (𝒫 𝑋 ∩ ∩ 𝑆) ∈ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗 ∈ 𝑆 𝑘 ⊆ 𝑗})
11		elssuni 4913	. . 3 ⊢ ((𝒫 𝑋 ∩ ∩ 𝑆) ∈ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗 ∈ 𝑆 𝑘 ⊆ 𝑗} → (𝒫 𝑋 ∩ ∩ 𝑆) ⊆ ∪ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗 ∈ 𝑆 𝑘 ⊆ 𝑗})
12	10, 11	syl 17	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → (𝒫 𝑋 ∩ ∩ 𝑆) ⊆ ∪ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗 ∈ 𝑆 𝑘 ⊆ 𝑗})
13		toponuni 22852	. . . . . . . . 9 ⊢ (𝑘 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝑘)
14		eqimss2 4018	. . . . . . . . 9 ⊢ (𝑋 = ∪ 𝑘 → ∪ 𝑘 ⊆ 𝑋)
15	13, 14	syl 17	. . . . . . . 8 ⊢ (𝑘 ∈ (TopOn‘𝑋) → ∪ 𝑘 ⊆ 𝑋)
16		sspwuni 5076	. . . . . . . 8 ⊢ (𝑘 ⊆ 𝒫 𝑋 ↔ ∪ 𝑘 ⊆ 𝑋)
17	15, 16	sylibr 234	. . . . . . 7 ⊢ (𝑘 ∈ (TopOn‘𝑋) → 𝑘 ⊆ 𝒫 𝑋)
18	17	3ad2ant2 1134	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) ∧ 𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗 ∈ 𝑆 𝑘 ⊆ 𝑗) → 𝑘 ⊆ 𝒫 𝑋)
19		simp3 1138	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) ∧ 𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗 ∈ 𝑆 𝑘 ⊆ 𝑗) → ∀𝑗 ∈ 𝑆 𝑘 ⊆ 𝑗)
20		ssint 4940	. . . . . . 7 ⊢ (𝑘 ⊆ ∩ 𝑆 ↔ ∀𝑗 ∈ 𝑆 𝑘 ⊆ 𝑗)
21	19, 20	sylibr 234	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) ∧ 𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗 ∈ 𝑆 𝑘 ⊆ 𝑗) → 𝑘 ⊆ ∩ 𝑆)
22	18, 21	ssind 4216	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) ∧ 𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗 ∈ 𝑆 𝑘 ⊆ 𝑗) → 𝑘 ⊆ (𝒫 𝑋 ∩ ∩ 𝑆))
23		velpw 4580	. . . . 5 ⊢ (𝑘 ∈ 𝒫 (𝒫 𝑋 ∩ ∩ 𝑆) ↔ 𝑘 ⊆ (𝒫 𝑋 ∩ ∩ 𝑆))
24	22, 23	sylibr 234	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) ∧ 𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗 ∈ 𝑆 𝑘 ⊆ 𝑗) → 𝑘 ∈ 𝒫 (𝒫 𝑋 ∩ ∩ 𝑆))
25	24	rabssdv 4050	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗 ∈ 𝑆 𝑘 ⊆ 𝑗} ⊆ 𝒫 (𝒫 𝑋 ∩ ∩ 𝑆))
26		sspwuni 5076	. . 3 ⊢ ({𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗 ∈ 𝑆 𝑘 ⊆ 𝑗} ⊆ 𝒫 (𝒫 𝑋 ∩ ∩ 𝑆) ↔ ∪ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗 ∈ 𝑆 𝑘 ⊆ 𝑗} ⊆ (𝒫 𝑋 ∩ ∩ 𝑆))
27	25, 26	sylib 218	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → ∪ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗 ∈ 𝑆 𝑘 ⊆ 𝑗} ⊆ (𝒫 𝑋 ∩ ∩ 𝑆))
28	12, 27	eqssd 3976	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → (𝒫 𝑋 ∩ ∩ 𝑆) = ∪ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗 ∈ 𝑆 𝑘 ⊆ 𝑗})

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108  ∀wral 3051  {crab 3415   ∩ cin 3925   ⊆ wss 3926  𝒫 cpw 4575  ∪ cuni 4883  ∩ cint 4922  ‘cfv 6531  TopOnctopon 22848

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-iota 6484  df-fun 6533  df-fv 6539  df-mre 17598  df-top 22832  df-topon 22849

This theorem is referenced by: (None)