Theorem topmeet 36560

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 36559	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → (𝒫 𝑋 ∩ ∩ 𝑆) ∈ (TopOn‘𝑋))
2		inss2 4191	. . . . . . 7 ⊢ (𝒫 𝑋 ∩ ∩ 𝑆) ⊆ ∩ 𝑆
3		intss1 4919	. . . . . . 7 ⊢ (𝑗 ∈ 𝑆 → ∩ 𝑆 ⊆ 𝑗)
4	2, 3	sstrid 3946	. . . . . 6 ⊢ (𝑗 ∈ 𝑆 → (𝒫 𝑋 ∩ ∩ 𝑆) ⊆ 𝑗)
5	4	rgen 3054	. . . . 5 ⊢ ∀𝑗 ∈ 𝑆 (𝒫 𝑋 ∩ ∩ 𝑆) ⊆ 𝑗
6		sseq1 3960	. . . . . . 7 ⊢ (𝑘 = (𝒫 𝑋 ∩ ∩ 𝑆) → (𝑘 ⊆ 𝑗 ↔ (𝒫 𝑋 ∩ ∩ 𝑆) ⊆ 𝑗))
7	6	ralbidv 3160	. . . . . 6 ⊢ (𝑘 = (𝒫 𝑋 ∩ ∩ 𝑆) → (∀𝑗 ∈ 𝑆 𝑘 ⊆ 𝑗 ↔ ∀𝑗 ∈ 𝑆 (𝒫 𝑋 ∩ ∩ 𝑆) ⊆ 𝑗))
8	7	elrab 3647	. . . . 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 711	. . . 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 4895	. . 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 22862	. . . . . . . . 9 ⊢ (𝑘 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝑘)
14		eqimss2 3994	. . . . . . . . 9 ⊢ (𝑋 = ∪ 𝑘 → ∪ 𝑘 ⊆ 𝑋)
15	13, 14	syl 17	. . . . . . . 8 ⊢ (𝑘 ∈ (TopOn‘𝑋) → ∪ 𝑘 ⊆ 𝑋)
16		sspwuni 5056	. . . . . . . 8 ⊢ (𝑘 ⊆ 𝒫 𝑋 ↔ ∪ 𝑘 ⊆ 𝑋)
17	15, 16	sylibr 234	. . . . . . 7 ⊢ (𝑘 ∈ (TopOn‘𝑋) → 𝑘 ⊆ 𝒫 𝑋)
18	17	3ad2ant2 1135	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) ∧ 𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗 ∈ 𝑆 𝑘 ⊆ 𝑗) → 𝑘 ⊆ 𝒫 𝑋)
19		simp3 1139	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) ∧ 𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗 ∈ 𝑆 𝑘 ⊆ 𝑗) → ∀𝑗 ∈ 𝑆 𝑘 ⊆ 𝑗)
20		ssint 4920	. . . . . . 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 4194	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) ∧ 𝑘 ∈ (TopOn‘𝑋) ∧ ∀𝑗 ∈ 𝑆 𝑘 ⊆ 𝑗) → 𝑘 ⊆ (𝒫 𝑋 ∩ ∩ 𝑆))
23		velpw 4560	. . . . 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 4027	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗 ∈ 𝑆 𝑘 ⊆ 𝑗} ⊆ 𝒫 (𝒫 𝑋 ∩ ∩ 𝑆))
26		sspwuni 5056	. . 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 3952	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑆 ⊆ (TopOn‘𝑋)) → (𝒫 𝑋 ∩ ∩ 𝑆) = ∪ {𝑘 ∈ (TopOn‘𝑋) ∣ ∀𝑗 ∈ 𝑆 𝑘 ⊆ 𝑗})

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  {crab 3400   ∩ cin 3901   ⊆ wss 3902  𝒫 cpw 4555  ∪ cuni 4864  ∩ cint 4903  ‘cfv 6493  TopOnctopon 22858

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4904  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-iota 6449  df-fun 6495  df-fv 6501  df-mre 17509  df-top 22842  df-topon 22859

This theorem is referenced by: (None)