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Theorem basis1 22316
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 22313 . . . 4 (𝐵 ∈ TopBases → (𝐵 ∈ TopBases ↔ ∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦))))
21ibi 267 . . 3 (𝐵 ∈ TopBases → ∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)))
3 ineq1 4170 . . . . 5 (𝑥 = 𝐶 → (𝑥𝑦) = (𝐶𝑦))
43pweqd 4582 . . . . . . 7 (𝑥 = 𝐶 → 𝒫 (𝑥𝑦) = 𝒫 (𝐶𝑦))
54ineq2d 4177 . . . . . 6 (𝑥 = 𝐶 → (𝐵 ∩ 𝒫 (𝑥𝑦)) = (𝐵 ∩ 𝒫 (𝐶𝑦)))
65unieqd 4884 . . . . 5 (𝑥 = 𝐶 (𝐵 ∩ 𝒫 (𝑥𝑦)) = (𝐵 ∩ 𝒫 (𝐶𝑦)))
73, 6sseq12d 3982 . . . 4 (𝑥 = 𝐶 → ((𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)) ↔ (𝐶𝑦) ⊆ (𝐵 ∩ 𝒫 (𝐶𝑦))))
8 ineq2 4171 . . . . 5 (𝑦 = 𝐷 → (𝐶𝑦) = (𝐶𝐷))
98pweqd 4582 . . . . . . 7 (𝑦 = 𝐷 → 𝒫 (𝐶𝑦) = 𝒫 (𝐶𝐷))
109ineq2d 4177 . . . . . 6 (𝑦 = 𝐷 → (𝐵 ∩ 𝒫 (𝐶𝑦)) = (𝐵 ∩ 𝒫 (𝐶𝐷)))
1110unieqd 4884 . . . . 5 (𝑦 = 𝐷 (𝐵 ∩ 𝒫 (𝐶𝑦)) = (𝐵 ∩ 𝒫 (𝐶𝐷)))
128, 11sseq12d 3982 . . . 4 (𝑦 = 𝐷 → ((𝐶𝑦) ⊆ (𝐵 ∩ 𝒫 (𝐶𝑦)) ↔ (𝐶𝐷) ⊆ (𝐵 ∩ 𝒫 (𝐶𝐷))))
137, 12rspc2v 3593 . . 3 ((𝐶𝐵𝐷𝐵) → (∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)) → (𝐶𝐷) ⊆ (𝐵 ∩ 𝒫 (𝐶𝐷))))
142, 13syl5com 31 . 2 (𝐵 ∈ TopBases → ((𝐶𝐵𝐷𝐵) → (𝐶𝐷) ⊆ (𝐵 ∩ 𝒫 (𝐶𝐷))))
15143impib 1117 1 ((𝐵 ∈ TopBases ∧ 𝐶𝐵𝐷𝐵) → (𝐶𝐷) ⊆ (𝐵 ∩ 𝒫 (𝐶𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  wral 3065  cin 3914  wss 3915  𝒫 cpw 4565   cuni 4870  TopBasesctb 22311
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-3an 1090  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rab 3411  df-v 3450  df-in 3922  df-ss 3932  df-pw 4567  df-uni 4871  df-bases 22312
This theorem is referenced by: (None)
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