Theorem isbasis2g 22098

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 22097	. 2 ⊢ (𝐵 ∈ 𝐶 → (𝐵 ∈ TopBases ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦))))
2		dfss3 3909	. . . 4 ⊢ ((𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)) ↔ ∀𝑧 ∈ (𝑥 ∩ 𝑦)𝑧 ∈ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)))
3		elin 3903	. . . . . . . . . 10 ⊢ (𝑤 ∈ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)) ↔ (𝑤 ∈ 𝐵 ∧ 𝑤 ∈ 𝒫 (𝑥 ∩ 𝑦)))
4		velpw 4538	. . . . . . . . . . 11 ⊢ (𝑤 ∈ 𝒫 (𝑥 ∩ 𝑦) ↔ 𝑤 ⊆ (𝑥 ∩ 𝑦))
5	4	anbi2i 623	. . . . . . . . . 10 ⊢ ((𝑤 ∈ 𝐵 ∧ 𝑤 ∈ 𝒫 (𝑥 ∩ 𝑦)) ↔ (𝑤 ∈ 𝐵 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦)))
6	3, 5	bitri 274	. . . . . . . . 9 ⊢ (𝑤 ∈ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)) ↔ (𝑤 ∈ 𝐵 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦)))
7	6	anbi2i 623	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦))) ↔ (𝑧 ∈ 𝑤 ∧ (𝑤 ∈ 𝐵 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦))))
8		an12 642	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝑤 ∧ (𝑤 ∈ 𝐵 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦))) ↔ (𝑤 ∈ 𝐵 ∧ (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦))))
9	7, 8	bitri 274	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦))) ↔ (𝑤 ∈ 𝐵 ∧ (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦))))
10	9	exbii 1850	. . . . . 6 ⊢ (∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦))) ↔ ∃𝑤(𝑤 ∈ 𝐵 ∧ (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦))))
11		eluni 4842	. . . . . 6 ⊢ (𝑧 ∈ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)) ↔ ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦))))
12		df-rex 3070	. . . . . 6 ⊢ (∃𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦)) ↔ ∃𝑤(𝑤 ∈ 𝐵 ∧ (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦))))
13	10, 11, 12	3bitr4i 303	. . . . 5 ⊢ (𝑧 ∈ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)) ↔ ∃𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦)))
14	13	ralbii 3092	. . . 4 ⊢ (∀𝑧 ∈ (𝑥 ∩ 𝑦)𝑧 ∈ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)) ↔ ∀𝑧 ∈ (𝑥 ∩ 𝑦)∃𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦)))
15	2, 14	bitri 274	. . 3 ⊢ ((𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)) ↔ ∀𝑧 ∈ (𝑥 ∩ 𝑦)∃𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦)))
16	15	2ralbii 3093	. 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 287	1 ⊢ (𝐵 ∈ 𝐶 → (𝐵 ∈ TopBases ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑥 ∩ 𝑦)∃𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ (𝑥 ∩ 𝑦))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396  ∃wex 1782   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065   ∩ cin 3886   ⊆ wss 3887  𝒫 cpw 4533  ∪ cuni 4839  TopBasesctb 22095

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-in 3894  df-ss 3904  df-pw 4535  df-uni 4840  df-bases 22096

This theorem is referenced by:  isbasis3g  22099  basis2  22101  fiinbas  22102  tgclb  22120  topbas  22122  restbas  22309  txbas  22718  blbas  23583