Theorem isbasis2g 23246

Description: Express the predicate "the set 𝐵 is a basis for a topology". (Contributed by NM, 17-Jul-2006.)

Assertion

Ref	Expression
isbasis2g	⊢ (𝐵 ∈ 𝐶 → (𝐵 ∈ TopBases ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑥 ∩ 𝑦)∃𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦))))

Distinct variable group:   𝑥,𝑤,𝑦,𝑧,𝐵

Allowed substitution hints:   𝐶(𝑥, 𝑦, 𝑧, 𝑤)

Proof of Theorem isbasis2g

Step	Hyp	Ref	Expression
1		isbasisg 23245	. 2 ⊢ (𝐵 ∈ 𝐶 → (𝐵 ∈ TopBases ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦))))
2		dfss3 3920	. . . 4 ⊢ ((𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)) ↔ ∀𝑧 ∈ (𝑥 ∩ 𝑦)𝑧 ∈ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)))
3		elin 3915	. . . . . . . . . 10 ⊢ (𝑤 ∈ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)) ↔ (𝑤 ∈ 𝐵 ∧ 𝑤 ∈ 𝒫 (𝑥 ∩ 𝑦)))
4		velpw 4562	. . . . . . . . . . 11 ⊢ (𝑤 ∈ 𝒫 (𝑥 ∩ 𝑦) ↔ 𝑤 ⊆ (𝑥 ∩ 𝑦))
5	4	anbi2i 635	. . . . . . . . . 10 ⊢ ((𝑤 ∈ 𝐵 ∧ 𝑤 ∈ 𝒫 (𝑥 ∩ 𝑦)) ↔ (𝑤 ∈ 𝐵 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦)))
6	3, 5	bitri 278	. . . . . . . . 9 ⊢ (𝑤 ∈ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)) ↔ (𝑤 ∈ 𝐵 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦)))
7	6	anbi2i 635	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦))) ↔ (𝑧 ∈ 𝑤 ∧ (𝑤 ∈ 𝐵 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦))))
8		an12 658	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑤 ∧ (𝑤 ∈ 𝐵 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦))) ↔ (𝑤 ∈ 𝐵 ∧ (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦))))
9	7, 8	bitri 278	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦))) ↔ (𝑤 ∈ 𝐵 ∧ (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦))))
10	9	exbii 1881	. . . . . 6 ⊢ (∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦))) ↔ ∃𝑤(𝑤 ∈ 𝐵 ∧ (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦))))
11		eluni 4870	. . . . . 6 ⊢ (𝑧 ∈ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)) ↔ ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦))))
12		df-rex 3088	. . . . . 6 ⊢ (∃𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦)) ↔ ∃𝑤(𝑤 ∈ 𝐵 ∧ (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦))))
13	10, 11, 12	3bitr4i 306	. . . . 5 ⊢ (𝑧 ∈ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)) ↔ ∃𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦)))
14	13	ralbii 3109	. . . 4 ⊢ (∀𝑧 ∈ (𝑥 ∩ 𝑦)𝑧 ∈ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)) ↔ ∀𝑧 ∈ (𝑥 ∩ 𝑦)∃𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦)))
15	2, 14	bitri 278	. . 3 ⊢ ((𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)) ↔ ∀𝑧 ∈ (𝑥 ∩ 𝑦)∃𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦)))
16	15	2ralbii 3138	. 2 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑥 ∩ 𝑦)∃𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦)))
17	1, 16	bitrdi 290	1 ⊢ (𝐵 ∈ 𝐶 → (𝐵 ∈ TopBases ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑥 ∩ 𝑦)∃𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∃wex 1812   ∈ wcel 2145  ∀wral 3077  ∃wrex 3087   ∩ cin 3898   ⊆ wss 3899  𝒫 cpw 4557  ∪ cuni 4867  TopBasesctb 23243

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-in 3906  df-ss 3916  df-pw 4559  df-uni 4868  df-bases 23244

This theorem is used by:  isbasis3g  23247  basis2  23249  fiinbas  23250  tgclb  23268  topbas  23270  restbas  23456  txbas  23866  blbas  24729