Theorem topmeet 36823

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 36822	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → (𝒫 𝑋 ∩ ∩ 𝑆) ∈ (TopOn‘𝑋))
2		inss2 4198	. . . . . . 7 ⊢ (𝒫 𝑋 ∩ ∩ 𝑆) ⊆ ∩ 𝑆
3		intss1 4933	. . . . . . 7 ⊢ (𝑗 ∈ 𝑆 → ∩ 𝑆 ⊆ 𝑗)
4	2, 3	sstrid 3956	. . . . . 6 ⊢ (𝑗 ∈ 𝑆 → (𝒫 𝑋 ∩ ∩ 𝑆) ⊆ 𝑗)
5	4	rgen 3088	. . . . 5 ⊢ ∀𝑗 ∈ 𝑆 (𝒫 𝑋 ∩ ∩ 𝑆) ⊆ 𝑗
6		sseq1 3970	. . . . . . 7 ⊢ (𝑘 = (𝒫 𝑋 ∩ ∩ 𝑆) → (𝑘 ⊆ 𝑗 ↔ (𝒫 𝑋 ∩ ∩ 𝑆) ⊆ 𝑗))
7	6	ralbidv 3195	. . . . . 6 ⊢ (𝑘 = (𝒫 𝑋 ∩ ∩ 𝑆) → (∀𝑗 ∈ 𝑆 𝑘 ⊆ 𝑗 ↔ ∀𝑗 ∈ 𝑆 (𝒫 𝑋 ∩ ∩ 𝑆) ⊆ 𝑗))
8	7	elrab 3658	. . . . 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 4909	. . 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 23054	. . . . . . . . 9 ⊢ (𝑘 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝑘)
14		eqimss2 4004	. . . . . . . . 9 ⊢ (𝑋 = ∪ 𝑘 → ∪ 𝑘 ⊆ 𝑋)
15	13, 14	syl 18	. . . . . . . 8 ⊢ (𝑘 ∈ (TopOn‘𝑋) → ∪ 𝑘 ⊆ 𝑋)
16		sspwuni 5071	. . . . . . . 8 ⊢ (𝑘 ⊆ 𝒫 𝑋 ↔ ∪ 𝑘 ⊆ 𝑋)
17	15, 16	sylibr 237	. . . . . . 7 ⊢ (𝑘 ∈ (TopOn‘𝑋) → 𝑘 ⊆ 𝒫 𝑋)
18	17	3ad2ant2 1150	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) ∧ 𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗 ∈ 𝑆 𝑘 ⊆ 𝑗) → 𝑘 ⊆ 𝒫 𝑋)
19		simp3 1154	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) ∧ 𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗 ∈ 𝑆 𝑘 ⊆ 𝑗) → ∀𝑗 ∈ 𝑆 𝑘 ⊆ 𝑗)
20		ssint 4934	. . . . . . 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 4201	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) ∧ 𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗 ∈ 𝑆 𝑘 ⊆ 𝑗) → 𝑘 ⊆ (𝒫 𝑋 ∩ ∩ 𝑆))
23		velpw 4572	. . . . 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 4036	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗 ∈ 𝑆 𝑘 ⊆ 𝑗} ⊆ 𝒫 (𝒫 𝑋 ∩ ∩ 𝑆))
26		sspwuni 5071	. . 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 3962	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → (𝒫 𝑋 ∩ ∩ 𝑆) = ∪ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗 ∈ 𝑆 𝑘 ⊆ 𝑗})

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1568   ∈ wcel 2150  ∀wral 3086  {crab 3423   ∩ cin 3912   ⊆ wss 3913  𝒫 cpw 4567  ∪ cuni 4877  ∩ cint 4917  ‘cfv 6540  TopOnctopon 23050

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-10 2183  ax-11 2199  ax-12 2220  ax-ext 2742  ax-sep 5262  ax-nul 5274  ax-pow 5340  ax-pr 5408  ax-un 7736

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2099  df-mo 2574  df-eu 2604  df-clab 2749  df-cleq 2762  df-clel 2845  df-nfc 2919  df-ne 2966  df-ral 3087  df-rex 3097  df-rab 3424  df-v 3464  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-int 4918  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5560  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-iota 6496  df-fun 6542  df-fv 6548  df-mre 17641  df-top 23034  df-topon 23051

This theorem is referenced by: (None)