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Theorem basis1 20565
 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 20562 . . . 4 (𝐵 ∈ TopBases → (𝐵 ∈ TopBases ↔ ∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦))))
21ibi 255 . . 3 (𝐵 ∈ TopBases → ∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)))
3 ineq1 3769 . . . . 5 (𝑥 = 𝐶 → (𝑥𝑦) = (𝐶𝑦))
43pweqd 4113 . . . . . . 7 (𝑥 = 𝐶 → 𝒫 (𝑥𝑦) = 𝒫 (𝐶𝑦))
54ineq2d 3776 . . . . . 6 (𝑥 = 𝐶 → (𝐵 ∩ 𝒫 (𝑥𝑦)) = (𝐵 ∩ 𝒫 (𝐶𝑦)))
65unieqd 4382 . . . . 5 (𝑥 = 𝐶 (𝐵 ∩ 𝒫 (𝑥𝑦)) = (𝐵 ∩ 𝒫 (𝐶𝑦)))
73, 6sseq12d 3597 . . . 4 (𝑥 = 𝐶 → ((𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)) ↔ (𝐶𝑦) ⊆ (𝐵 ∩ 𝒫 (𝐶𝑦))))
8 ineq2 3770 . . . . 5 (𝑦 = 𝐷 → (𝐶𝑦) = (𝐶𝐷))
98pweqd 4113 . . . . . . 7 (𝑦 = 𝐷 → 𝒫 (𝐶𝑦) = 𝒫 (𝐶𝐷))
109ineq2d 3776 . . . . . 6 (𝑦 = 𝐷 → (𝐵 ∩ 𝒫 (𝐶𝑦)) = (𝐵 ∩ 𝒫 (𝐶𝐷)))
1110unieqd 4382 . . . . 5 (𝑦 = 𝐷 (𝐵 ∩ 𝒫 (𝐶𝑦)) = (𝐵 ∩ 𝒫 (𝐶𝐷)))
128, 11sseq12d 3597 . . . 4 (𝑦 = 𝐷 → ((𝐶𝑦) ⊆ (𝐵 ∩ 𝒫 (𝐶𝑦)) ↔ (𝐶𝐷) ⊆ (𝐵 ∩ 𝒫 (𝐶𝐷))))
137, 12rspc2v 3293 . . 3 ((𝐶𝐵𝐷𝐵) → (∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)) → (𝐶𝐷) ⊆ (𝐵 ∩ 𝒫 (𝐶𝐷))))
142, 13syl5com 31 . 2 (𝐵 ∈ TopBases → ((𝐶𝐵𝐷𝐵) → (𝐶𝐷) ⊆ (𝐵 ∩ 𝒫 (𝐶𝐷))))
15143impib 1254 1 ((𝐵 ∈ TopBases ∧ 𝐶𝐵𝐷𝐵) → (𝐶𝐷) ⊆ (𝐵 ∩ 𝒫 (𝐶𝐷)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ∩ cin 3539   ⊆ wss 3540  𝒫 cpw 4108  ∪ cuni 4372  TopBasesctb 20520 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-in 3547  df-ss 3554  df-pw 4110  df-uni 4373  df-bases 20522 This theorem is referenced by: (None)
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