Theorem basis1 14283

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 14280	. . . 4 ⊢ (𝐵 ∈ TopBases → (𝐵 ∈ TopBases ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦))))
2	1	ibi 176	. . 3 ⊢ (𝐵 ∈ TopBases → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)))
3		ineq1 3357	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝑥 ∩ 𝑦) = (𝐶 ∩ 𝑦))
4	3	pweqd 3610	. . . . . . 7 ⊢ (𝑥 = 𝐶 → 𝒫 (𝑥 ∩ 𝑦) = 𝒫 (𝐶 ∩ 𝑦))
5	4	ineq2d 3364	. . . . . 6 ⊢ (𝑥 = 𝐶 → (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)) = (𝐵 ∩ 𝒫 (𝐶 ∩ 𝑦)))
6	5	unieqd 3850	. . . . 5 ⊢ (𝑥 = 𝐶 → ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)) = ∪ (𝐵 ∩ 𝒫 (𝐶 ∩ 𝑦)))
7	3, 6	sseq12d 3214	. . . 4 ⊢ (𝑥 = 𝐶 → ((𝑥 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝑥 ∩ 𝑦)) ↔ (𝐶 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝐶 ∩ 𝑦))))
8		ineq2 3358	. . . . 5 ⊢ (𝑦 = 𝐷 → (𝐶 ∩ 𝑦) = (𝐶 ∩ 𝐷))
9	8	pweqd 3610	. . . . . . 7 ⊢ (𝑦 = 𝐷 → 𝒫 (𝐶 ∩ 𝑦) = 𝒫 (𝐶 ∩ 𝐷))
10	9	ineq2d 3364	. . . . . 6 ⊢ (𝑦 = 𝐷 → (𝐵 ∩ 𝒫 (𝐶 ∩ 𝑦)) = (𝐵 ∩ 𝒫 (𝐶 ∩ 𝐷)))
11	10	unieqd 3850	. . . . 5 ⊢ (𝑦 = 𝐷 → ∪ (𝐵 ∩ 𝒫 (𝐶 ∩ 𝑦)) = ∪ (𝐵 ∩ 𝒫 (𝐶 ∩ 𝐷)))
12	8, 11	sseq12d 3214	. . . 4 ⊢ (𝑦 = 𝐷 → ((𝐶 ∩ 𝑦) ⊆ ∪ (𝐵 ∩ 𝒫 (𝐶 ∩ 𝑦)) ↔ (𝐶 ∩ 𝐷) ⊆ ∪ (𝐵 ∩ 𝒫 (𝐶 ∩ 𝐷))))
13	7, 12	rspc2v 2881	. . 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 1203	1 ⊢ ((𝐵 ∈ TopBases ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵) → (𝐶 ∩ 𝐷) ⊆ ∪ (𝐵 ∩ 𝒫 (𝐶 ∩ 𝐷)))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 980   = wceq 1364   ∈ wcel 2167  ∀wral 2475   ∩ cin 3156   ⊆ wss 3157  𝒫 cpw 3605  ∪ cuni 3839  TopBasesctb 14278

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-in 3163  df-ss 3170  df-pw 3607  df-uni 3840  df-bases 14279

This theorem is referenced by: (None)