Theorem isbasis2g 23086

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 23085	. 2 ⊢ (𝐵 ∈ 𝐶 → (𝐵 ∈ TopBases ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦))))
2		dfss3 3927	. . . 4 ⊢ ((𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)) ↔ ∀𝑧 ∈ (𝑥 ∩ 𝑦)𝑧 ∈ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)))
3		elin 3922	. . . . . . . . . 10 ⊢ (𝑤 ∈ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)) ↔ (𝑤 ∈ 𝐵 ∧ 𝑤 ∈ 𝒫 (𝑥 ∩ 𝑦)))
4		velpw 4568	. . . . . . . . . . 11 ⊢ (𝑤 ∈ 𝒫 (𝑥 ∩ 𝑦) ↔ 𝑤 ⊆ (𝑥 ∩ 𝑦))
5	4	anbi2i 634	. . . . . . . . . 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 634	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦))) ↔ (𝑧 ∈ 𝑤 ∧ (𝑤 ∈ 𝐵 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦))))
8		an12 657	. . . . . . . 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 1878	. . . . . 6 ⊢ (∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦))) ↔ ∃𝑤(𝑤 ∈ 𝐵 ∧ (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦))))
11		eluni 4876	. . . . . 6 ⊢ (𝑧 ∈ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)) ↔ ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦))))
12		df-rex 3090	. . . . . 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 3111	. . . 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 3140	. 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))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∃wex 1809   ∈ wcel 2143  ∀wral 3079  ∃wrex 3089   ∩ cin 3905   ⊆ wss 3906  𝒫 cpw 4563  ∪ cuni 4873  TopBasesctb 23083

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-in 3913  df-ss 3923  df-pw 4565  df-uni 4874  df-bases 23084

This theorem is referenced by:  isbasis3g  23087  basis2  23089  fiinbas  23090  tgclb  23108  topbas  23110  restbas  23296  txbas  23705  blbas  24568