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Theorem basis1 21650
 Description: Property of a basis. (Contributed by NM, 16-Jul-2006.)
Assertion
Ref Expression
basis1 ((𝐵 ∈ TopBases ∧ 𝐶𝐵𝐷𝐵) → (𝐶𝐷) ⊆ (𝐵 ∩ 𝒫 (𝐶𝐷)))

Proof of Theorem basis1
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isbasisg 21647 . . . 4 (𝐵 ∈ TopBases → (𝐵 ∈ TopBases ↔ ∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦))))
21ibi 270 . . 3 (𝐵 ∈ TopBases → ∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)))
3 ineq1 4109 . . . . 5 (𝑥 = 𝐶 → (𝑥𝑦) = (𝐶𝑦))
43pweqd 4513 . . . . . . 7 (𝑥 = 𝐶 → 𝒫 (𝑥𝑦) = 𝒫 (𝐶𝑦))
54ineq2d 4117 . . . . . 6 (𝑥 = 𝐶 → (𝐵 ∩ 𝒫 (𝑥𝑦)) = (𝐵 ∩ 𝒫 (𝐶𝑦)))
65unieqd 4812 . . . . 5 (𝑥 = 𝐶 (𝐵 ∩ 𝒫 (𝑥𝑦)) = (𝐵 ∩ 𝒫 (𝐶𝑦)))
73, 6sseq12d 3925 . . . 4 (𝑥 = 𝐶 → ((𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)) ↔ (𝐶𝑦) ⊆ (𝐵 ∩ 𝒫 (𝐶𝑦))))
8 ineq2 4111 . . . . 5 (𝑦 = 𝐷 → (𝐶𝑦) = (𝐶𝐷))
98pweqd 4513 . . . . . . 7 (𝑦 = 𝐷 → 𝒫 (𝐶𝑦) = 𝒫 (𝐶𝐷))
109ineq2d 4117 . . . . . 6 (𝑦 = 𝐷 → (𝐵 ∩ 𝒫 (𝐶𝑦)) = (𝐵 ∩ 𝒫 (𝐶𝐷)))
1110unieqd 4812 . . . . 5 (𝑦 = 𝐷 (𝐵 ∩ 𝒫 (𝐶𝑦)) = (𝐵 ∩ 𝒫 (𝐶𝐷)))
128, 11sseq12d 3925 . . . 4 (𝑦 = 𝐷 → ((𝐶𝑦) ⊆ (𝐵 ∩ 𝒫 (𝐶𝑦)) ↔ (𝐶𝐷) ⊆ (𝐵 ∩ 𝒫 (𝐶𝐷))))
137, 12rspc2v 3551 . . 3 ((𝐶𝐵𝐷𝐵) → (∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)) → (𝐶𝐷) ⊆ (𝐵 ∩ 𝒫 (𝐶𝐷))))
142, 13syl5com 31 . 2 (𝐵 ∈ TopBases → ((𝐶𝐵𝐷𝐵) → (𝐶𝐷) ⊆ (𝐵 ∩ 𝒫 (𝐶𝐷))))
15143impib 1113 1 ((𝐵 ∈ TopBases ∧ 𝐶𝐵𝐷𝐵) → (𝐶𝐷) ⊆ (𝐵 ∩ 𝒫 (𝐶𝐷)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3070   ∩ cin 3857   ⊆ wss 3858  𝒫 cpw 4494  ∪ cuni 4798  TopBasesctb 21645 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 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rab 3079  df-v 3411  df-in 3865  df-ss 3875  df-pw 4496  df-uni 4799  df-bases 21646 This theorem is referenced by: (None)
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