Theorem isbasisg 12200

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

Assertion

Ref	Expression
isbasisg	⊢ (𝐵 ∈ 𝐶 → (𝐵 ∈ TopBases ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦))))

Distinct variable group:   𝑥,𝑦,𝐵

Allowed substitution hints:   𝐶(𝑥,𝑦)

Proof of Theorem isbasisg

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ineq1 3265	. . . . . 6 ⊢ (𝑧 = 𝐵 → (𝑧 ∩ 𝒫 (𝑥 ∩ 𝑦)) = (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)))
2	1	unieqd 3742	. . . . 5 ⊢ (𝑧 = 𝐵 → ∪ (𝑧 ∩ 𝒫 (𝑥 ∩ 𝑦)) = ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)))
3	2	sseq2d 3122	. . . 4 ⊢ (𝑧 = 𝐵 → ((𝑥 ∩ 𝑦) ⊆ ∪ (𝑧 ∩ 𝒫 (𝑥 ∩ 𝑦)) ↔ (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦))))
4	3	raleqbi1dv 2632	. . 3 ⊢ (𝑧 = 𝐵 → (∀𝑦 ∈ 𝑧 (𝑥 ∩ 𝑦) ⊆ ∪ (𝑧 ∩ 𝒫 (𝑥 ∩ 𝑦)) ↔ ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦))))
5	4	raleqbi1dv 2632	. 2 ⊢ (𝑧 = 𝐵 → (∀𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 (𝑥 ∩ 𝑦) ⊆ ∪ (𝑧 ∩ 𝒫 (𝑥 ∩ 𝑦)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦))))
6		df-bases 12199	. 2 ⊢ TopBases = {𝑧 ∣ ∀𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 (𝑥 ∩ 𝑦) ⊆ ∪ (𝑧 ∩ 𝒫 (𝑥 ∩ 𝑦))}
7	5, 6	elab2g 2826	1 ⊢ (𝐵 ∈ 𝐶 → (𝐵 ∈ TopBases ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦))))

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1331   ∈ wcel 1480  ∀wral 2414   ∩ cin 3065   ⊆ wss 3066  𝒫 cpw 3505  ∪ cuni 3731  TopBasesctb 12198

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-in 3072  df-ss 3079  df-uni 3732  df-bases 12199

This theorem is referenced by:  isbasis2g  12201  basis1  12203  baspartn  12206