Theorem isbasisg 14958

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 3417	. . . . . 6 ⊢ (𝑧 = 𝐵 → (𝑧 ∩ 𝒫 (𝑥 ∩ 𝑦)) = (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)))
2	1	unieqd 3927	. . . . 5 ⊢ (𝑧 = 𝐵 → ∪ (𝑧 ∩ 𝒫 (𝑥 ∩ 𝑦)) = ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)))
3	2	sseq2d 3270	. . . 4 ⊢ (𝑧 = 𝐵 → ((𝑥 ∩ 𝑦) ⊆ ∪ (𝑧 ∩ 𝒫 (𝑥 ∩ 𝑦)) ↔ (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦))))
4	3	raleqbi1dv 2755	. . 3 ⊢ (𝑧 = 𝐵 → (∀𝑦 ∈ 𝑧 (𝑥 ∩ 𝑦) ⊆ ∪ (𝑧 ∩ 𝒫 (𝑥 ∩ 𝑦)) ↔ ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦))))
5	4	raleqbi1dv 2755	. 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 14957	. 2 ⊢ TopBases = {𝑧 ∣ ∀𝑥 ∈ 𝑧 ∀𝑦 ∈ 𝑧 (𝑥 ∩ 𝑦) ⊆ ∪ (𝑧 ∩ 𝒫 (𝑥 ∩ 𝑦))}
7	5, 6	elab2g 2966	1 ⊢ (𝐵 ∈ 𝐶 → (𝐵 ∈ TopBases ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦))))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1398   ∈ wcel 2205  ∀wral 2522   ∩ cin 3212   ⊆ wss 3213  𝒫 cpw 3671  ∪ cuni 3916  TopBasesctb 14956

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-in 3219  df-ss 3226  df-uni 3917  df-bases 14957

This theorem is referenced by:  isbasis2g  14959  basis1  14961  baspartn  14964