Theorem isbasisg 21077

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 4004	. . . . . 6 ⊢ (𝑧 = 𝐵 → (𝑧 ∩ 𝒫 (𝑥 ∩ 𝑦)) = (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)))
2	1	unieqd 4637	. . . . 5 ⊢ (𝑧 = 𝐵 → ∪ (𝑧 ∩ 𝒫 (𝑥 ∩ 𝑦)) = ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)))
3	2	sseq2d 3828	. . . 4 ⊢ (𝑧 = 𝐵 → ((𝑥 ∩ 𝑦) ⊆ ∪ (𝑧 ∩ 𝒫 (𝑥 ∩ 𝑦)) ↔ (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦))))
4	3	raleqbi1dv 3328	. . 3 ⊢ (𝑧 = 𝐵 → (∀𝑦 ∈ 𝑧 (𝑥 ∩ 𝑦) ⊆ ∪ (𝑧 ∩ 𝒫 (𝑥 ∩ 𝑦)) ↔ ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦))))
5	4	raleqbi1dv 3328	. 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 21076	. 2 ⊢ TopBases = {𝑧 ∣ ∀𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 (𝑥 ∩ 𝑦) ⊆ ∪ (𝑧 ∩ 𝒫 (𝑥 ∩ 𝑦))}
7	5, 6	elab2g 3544	1 ⊢ (𝐵 ∈ 𝐶 → (𝐵 ∈ TopBases ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦))))

Syntax hints:   → wi 4   ↔ wb 198   = wceq 1653   ∈ wcel 2157  ∀wral 3088   ∩ cin 3767   ⊆ wss 3768  𝒫 cpw 4348  ∪ cuni 4627  TopBasesctb 21075

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2776

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2785  df-cleq 2791  df-clel 2794  df-nfc 2929  df-ral 3093  df-rex 3094  df-v 3386  df-in 3775  df-ss 3782  df-uni 4628  df-bases 21076

This theorem is referenced by:  isbasis2g  21078  basis1  21080  basdif0  21083  baspartn  21084  basqtop  21840