Theorem isbasisg 21552

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 4131	. . . . . 6 ⊢ (𝑧 = 𝐵 → (𝑧 ∩ 𝒫 (𝑥 ∩ 𝑦)) = (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)))
2	1	unieqd 4814	. . . . 5 ⊢ (𝑧 = 𝐵 → ∪ (𝑧 ∩ 𝒫 (𝑥 ∩ 𝑦)) = ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)))
3	2	sseq2d 3947	. . . 4 ⊢ (𝑧 = 𝐵 → ((𝑥 ∩ 𝑦) ⊆ ∪ (𝑧 ∩ 𝒫 (𝑥 ∩ 𝑦)) ↔ (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦))))
4	3	raleqbi1dv 3356	. . 3 ⊢ (𝑧 = 𝐵 → (∀𝑦 ∈ 𝑧 (𝑥 ∩ 𝑦) ⊆ ∪ (𝑧 ∩ 𝒫 (𝑥 ∩ 𝑦)) ↔ ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦))))
5	4	raleqbi1dv 3356	. 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 21551	. 2 ⊢ TopBases = {𝑧 ∣ ∀𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 (𝑥 ∩ 𝑦) ⊆ ∪ (𝑧 ∩ 𝒫 (𝑥 ∩ 𝑦))}
7	5, 6	elab2g 3616	1 ⊢ (𝐵 ∈ 𝐶 → (𝐵 ∈ TopBases ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦))))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538   ∈ wcel 2111  ∀wral 3106   ∩ cin 3880   ⊆ wss 3881  𝒫 cpw 4497  ∪ cuni 4800  TopBasesctb 21550

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rab 3115  df-v 3443  df-in 3888  df-ss 3898  df-uni 4801  df-bases 21551

This theorem is referenced by:  isbasis2g  21553  basis1  21555  basdif0  21558  baspartn  21559  basqtop  22316