Theorem topmeet 36798

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 36797	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → (𝒫 𝑋 ∩ ∩ 𝑆) ∈ (TopOn‘𝑋))
2		inss2 4196	. . . . . . 7 ⊢ (𝒫 𝑋 ∩ ∩ 𝑆) ⊆ ∩ 𝑆
3		intss1 4930	. . . . . . 7 ⊢ (𝑗 ∈ 𝑆 → ∩ 𝑆 ⊆ 𝑗)
4	2, 3	sstrid 3954	. . . . . 6 ⊢ (𝑗 ∈ 𝑆 → (𝒫 𝑋 ∩ ∩ 𝑆) ⊆ 𝑗)
5	4	rgen 3087	. . . . 5 ⊢ ∀𝑗 ∈ 𝑆 (𝒫 𝑋 ∩ ∩ 𝑆) ⊆ 𝑗
6		sseq1 3968	. . . . . . 7 ⊢ (𝑘 = (𝒫 𝑋 ∩ ∩ 𝑆) → (𝑘 ⊆ 𝑗 ↔ (𝒫 𝑋 ∩ ∩ 𝑆) ⊆ 𝑗))
7	6	ralbidv 3194	. . . . . 6 ⊢ (𝑘 = (𝒫 𝑋 ∩ ∩ 𝑆) → (∀𝑗 ∈ 𝑆 𝑘 ⊆ 𝑗 ↔ ∀𝑗 ∈ 𝑆 (𝒫 𝑋 ∩ ∩ 𝑆) ⊆ 𝑗))
8	7	elrab 3657	. . . . 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 4906	. . 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 23040	. . . . . . . . 9 ⊢ (𝑘 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝑘)
14		eqimss2 4002	. . . . . . . . 9 ⊢ (𝑋 = ∪ 𝑘 → ∪ 𝑘 ⊆ 𝑋)
15	13, 14	syl 18	. . . . . . . 8 ⊢ (𝑘 ∈ (TopOn‘𝑋) → ∪ 𝑘 ⊆ 𝑋)
16		sspwuni 5068	. . . . . . . 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 4931	. . . . . . 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 4199	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) ∧ 𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗 ∈ 𝑆 𝑘 ⊆ 𝑗) → 𝑘 ⊆ (𝒫 𝑋 ∩ ∩ 𝑆))
23		velpw 4570	. . . . 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 4034	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗 ∈ 𝑆 𝑘 ⊆ 𝑗} ⊆ 𝒫 (𝒫 𝑋 ∩ ∩ 𝑆))
26		sspwuni 5068	. . 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 3960	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 1567   ∈ wcel 2149  ∀wral 3085  {crab 3422   ∩ cin 3910   ⊆ wss 3911  𝒫 cpw 4565  ∪ cuni 4874  ∩ cint 4914  ‘cfv 6537  TopOnctopon 23036

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5259  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3423  df-v 3463  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4491  df-pw 4567  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4915  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-iota 6493  df-fun 6539  df-fv 6545  df-mre 17638  df-top 23020  df-topon 23037

This theorem is referenced by: (None)