Theorem basis1 23248

Description: Property of a basis. (Contributed by NM, 16-Jul-2006.)

Assertion

Ref	Expression
basis1	⊢ ((𝐵 ∈ TopBases ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵) → (𝐶 ∩ 𝐷) ⊆ ∪ (𝐵 ∩ 𝒫 (𝐶 ∩ 𝐷)))

Proof of Theorem basis1

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isbasisg 23245	. . . 4 ⊢ (𝐵 ∈ TopBases → (𝐵 ∈ TopBases ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦))))
2	1	ibi 270	. . 3 ⊢ (𝐵 ∈ TopBases → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)))
3		ineq1 4159	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝑥 ∩ 𝑦) = (𝐶 ∩ 𝑦))
4	3	pweqd 4574	. . . . . . 7 ⊢ (𝑥 = 𝐶 → 𝒫 (𝑥 ∩ 𝑦) = 𝒫 (𝐶 ∩ 𝑦))
5	4	ineq2d 4166	. . . . . 6 ⊢ (𝑥 = 𝐶 → (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)) = (𝐵 ∩ 𝒫 (𝐶 ∩ 𝑦)))
6	5	unieqd 4880	. . . . 5 ⊢ (𝑥 = 𝐶 → ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)) = ∪ (𝐵 ∩ 𝒫 (𝐶 ∩ 𝑦)))
7	3, 6	sseq12d 3964	. . . 4 ⊢ (𝑥 = 𝐶 → ((𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)) ↔ (𝐶 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝐶 ∩ 𝑦))))
8		ineq2 4160	. . . . 5 ⊢ (𝑦 = 𝐷 → (𝐶 ∩ 𝑦) = (𝐶 ∩ 𝐷))
9	8	pweqd 4574	. . . . . . 7 ⊢ (𝑦 = 𝐷 → 𝒫 (𝐶 ∩ 𝑦) = 𝒫 (𝐶 ∩ 𝐷))
10	9	ineq2d 4166	. . . . . 6 ⊢ (𝑦 = 𝐷 → (𝐵 ∩ 𝒫 (𝐶 ∩ 𝑦)) = (𝐵 ∩ 𝒫 (𝐶 ∩ 𝐷)))
11	10	unieqd 4880	. . . . 5 ⊢ (𝑦 = 𝐷 → ∪ (𝐵 ∩ 𝒫 (𝐶 ∩ 𝑦)) = ∪ (𝐵 ∩ 𝒫 (𝐶 ∩ 𝐷)))
12	8, 11	sseq12d 3964	. . . 4 ⊢ (𝑦 = 𝐷 → ((𝐶 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝐶 ∩ 𝑦)) ↔ (𝐶 ∩ 𝐷) ⊆ ∪ (𝐵 ∩ 𝒫 (𝐶 ∩ 𝐷))))
13	7, 12	rspc2v 3587	. . 3 ⊢ ((𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)) → (𝐶 ∩ 𝐷) ⊆ ∪ (𝐵 ∩ 𝒫 (𝐶 ∩ 𝐷))))
14	2, 13	syl5com 32	. 2 ⊢ (𝐵 ∈ TopBases → ((𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵) → (𝐶 ∩ 𝐷) ⊆ ∪ (𝐵 ∩ 𝒫 (𝐶 ∩ 𝐷))))
15	14	3impib 1134	1 ⊢ ((𝐵 ∈ TopBases ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵) → (𝐶 ∩ 𝐷) ⊆ ∪ (𝐵 ∩ 𝒫 (𝐶 ∩ 𝐷)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3077   ∩ cin 3898   ⊆ wss 3899  𝒫 cpw 4557  ∪ cuni 4867  TopBasesctb 23243

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-in 3906  df-ss 3916  df-pw 4559  df-uni 4868  df-bases 23244

This theorem is used by: (None)