Theorem basis1 12219

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 12216	. . . 4 ⊢ (𝐵 ∈ TopBases → (𝐵 ∈ TopBases ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦))))
2	1	ibi 175	. . 3 ⊢ (𝐵 ∈ TopBases → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)))
3		ineq1 3270	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝑥 ∩ 𝑦) = (𝐶 ∩ 𝑦))
4	3	pweqd 3515	. . . . . . 7 ⊢ (𝑥 = 𝐶 → 𝒫 (𝑥 ∩ 𝑦) = 𝒫 (𝐶 ∩ 𝑦))
5	4	ineq2d 3277	. . . . . 6 ⊢ (𝑥 = 𝐶 → (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)) = (𝐵 ∩ 𝒫 (𝐶 ∩ 𝑦)))
6	5	unieqd 3747	. . . . 5 ⊢ (𝑥 = 𝐶 → ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)) = ∪ (𝐵 ∩ 𝒫 (𝐶 ∩ 𝑦)))
7	3, 6	sseq12d 3128	. . . 4 ⊢ (𝑥 = 𝐶 → ((𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)) ↔ (𝐶 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝐶 ∩ 𝑦))))
8		ineq2 3271	. . . . 5 ⊢ (𝑦 = 𝐷 → (𝐶 ∩ 𝑦) = (𝐶 ∩ 𝐷))
9	8	pweqd 3515	. . . . . . 7 ⊢ (𝑦 = 𝐷 → 𝒫 (𝐶 ∩ 𝑦) = 𝒫 (𝐶 ∩ 𝐷))
10	9	ineq2d 3277	. . . . . 6 ⊢ (𝑦 = 𝐷 → (𝐵 ∩ 𝒫 (𝐶 ∩ 𝑦)) = (𝐵 ∩ 𝒫 (𝐶 ∩ 𝐷)))
11	10	unieqd 3747	. . . . 5 ⊢ (𝑦 = 𝐷 → ∪ (𝐵 ∩ 𝒫 (𝐶 ∩ 𝑦)) = ∪ (𝐵 ∩ 𝒫 (𝐶 ∩ 𝐷)))
12	8, 11	sseq12d 3128	. . . 4 ⊢ (𝑦 = 𝐷 → ((𝐶 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝐶 ∩ 𝑦)) ↔ (𝐶 ∩ 𝐷) ⊆ ∪ (𝐵 ∩ 𝒫 (𝐶 ∩ 𝐷))))
13	7, 12	rspc2v 2802	. . 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 29	. 2 ⊢ (𝐵 ∈ TopBases → ((𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵) → (𝐶 ∩ 𝐷) ⊆ ∪ (𝐵 ∩ 𝒫 (𝐶 ∩ 𝐷))))
15	14	3impib 1179	1 ⊢ ((𝐵 ∈ TopBases ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵) → (𝐶 ∩ 𝐷) ⊆ ∪ (𝐵 ∩ 𝒫 (𝐶 ∩ 𝐷)))

Syntax hints:   → wi 4   ∧ wa 103   ∧ w3a 962   = wceq 1331   ∈ wcel 1480  ∀wral 2416   ∩ cin 3070   ⊆ wss 3071  𝒫 cpw 3510  ∪ cuni 3736  TopBasesctb 12214

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 2121

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-in 3077  df-ss 3084  df-pw 3512  df-uni 3737  df-bases 12215

This theorem is referenced by: (None)