Theorem topmeet 36903

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 36902	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → (𝒫 𝑋 ∩ ∩ 𝑆) ∈ (TopOn‘𝑋))
2		inss2 4189	. . . . . . 7 ⊢ (𝒫 𝑋 ∩ ∩ 𝑆) ⊆ ∩ 𝑆
3		intss1 4927	. . . . . . 7 ⊢ (𝑗 ∈ 𝑆 → ∩ 𝑆 ⊆ 𝑗)
4	2, 3	sstrid 3947	. . . . . 6 ⊢ (𝑗 ∈ 𝑆 → (𝒫 𝑋 ∩ ∩ 𝑆) ⊆ 𝑗)
5	4	rgen 3080	. . . . 5 ⊢ ∀𝑗 ∈ 𝑆 (𝒫 𝑋 ∩ ∩ 𝑆) ⊆ 𝑗
6		sseq1 3961	. . . . . . 7 ⊢ (𝑘 = (𝒫 𝑋 ∩ ∩ 𝑆) → (𝑘 ⊆ 𝑗 ↔ (𝒫 𝑋 ∩ ∩ 𝑆) ⊆ 𝑗))
7	6	ralbidv 3187	. . . . . 6 ⊢ (𝑘 = (𝒫 𝑋 ∩ ∩ 𝑆) → (∀𝑗 ∈ 𝑆 𝑘 ⊆ 𝑗 ↔ ∀𝑗 ∈ 𝑆 (𝒫 𝑋 ∩ ∩ 𝑆) ⊆ 𝑗))
8	7	elrab 3649	. . . . 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 722	. . . 4 ⊢ ((𝒫 𝑋 ∩ ∩ 𝑆) ∈ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗 ∈ 𝑆 𝑘 ⊆ 𝑗} ↔ (𝒫 𝑋 ∩ ∩ 𝑆) ∈ (TopOn‘𝑋))
10	1, 9	sylibr 237	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → (𝒫 𝑋 ∩ ∩ 𝑆) ∈ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗 ∈ 𝑆 𝑘 ⊆ 𝑗})
11		elssuni 4903	. . 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 18	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → (𝒫 𝑋 ∩ ∩ 𝑆) ⊆ ∪ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗 ∈ 𝑆 𝑘 ⊆ 𝑗})
13		toponuni 23082	. . . . . . . . 9 ⊢ (𝑘 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝑘)
14		eqimss2 3995	. . . . . . . . 9 ⊢ (𝑋 = ∪ 𝑘 → ∪ 𝑘 ⊆ 𝑋)
15	13, 14	syl 18	. . . . . . . 8 ⊢ (𝑘 ∈ (TopOn‘𝑋) → ∪ 𝑘 ⊆ 𝑋)
16		sspwuni 5065	. . . . . . . 8 ⊢ (𝑘 ⊆ 𝒫 𝑋 ↔ ∪ 𝑘 ⊆ 𝑋)
17	15, 16	sylibr 237	. . . . . . 7 ⊢ (𝑘 ∈ (TopOn‘𝑋) → 𝑘 ⊆ 𝒫 𝑋)
18	17	3ad2ant2 1151	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) ∧ 𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗 ∈ 𝑆 𝑘 ⊆ 𝑗) → 𝑘 ⊆ 𝒫 𝑋)
19		simp3 1155	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) ∧ 𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗 ∈ 𝑆 𝑘 ⊆ 𝑗) → ∀𝑗 ∈ 𝑆 𝑘 ⊆ 𝑗)
20		ssint 4928	. . . . . . 7 ⊢ (𝑘 ⊆ ∩ 𝑆 ↔ ∀𝑗 ∈ 𝑆 𝑘 ⊆ 𝑗)
21	19, 20	sylibr 237	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) ∧ 𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗 ∈ 𝑆 𝑘 ⊆ 𝑗) → 𝑘 ⊆ ∩ 𝑆)
22	18, 21	ssind 4192	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) ∧ 𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗 ∈ 𝑆 𝑘 ⊆ 𝑗) → 𝑘 ⊆ (𝒫 𝑋 ∩ ∩ 𝑆))
23		velpw 4566	. . . . 5 ⊢ (𝑘 ∈ 𝒫 (𝒫 𝑋 ∩ ∩ 𝑆) ↔ 𝑘 ⊆ (𝒫 𝑋 ∩ ∩ 𝑆))
24	22, 23	sylibr 237	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) ∧ 𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗 ∈ 𝑆 𝑘 ⊆ 𝑗) → 𝑘 ∈ 𝒫 (𝒫 𝑋 ∩ ∩ 𝑆))
25	24	rabssdv 4027	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗 ∈ 𝑆 𝑘 ⊆ 𝑗} ⊆ 𝒫 (𝒫 𝑋 ∩ ∩ 𝑆))
26		sspwuni 5065	. . 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 221	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → ∪ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗 ∈ 𝑆 𝑘 ⊆ 𝑗} ⊆ (𝒫 𝑋 ∩ ∩ 𝑆))
28	12, 27	eqssd 3953	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → (𝒫 𝑋 ∩ ∩ 𝑆) = ∪ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗 ∈ 𝑆 𝑘 ⊆ 𝑗})

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   ∧ w3a 1102   = wceq 1569   ∈ wcel 2142  ∀wral 3078  {crab 3415   ∩ cin 3903   ⊆ wss 3904  𝒫 cpw 4561  ∪ cuni 4871  ∩ cint 4911  ‘cfv 6536  TopOnctopon 23078

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-int 4912  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-iota 6492  df-fun 6538  df-fv 6544  df-mre 17644  df-top 23062  df-topon 23079

This theorem is used by: (None)