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Theorem basis1 12141
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 12138 . . . 4 (𝐵 ∈ TopBases → (𝐵 ∈ TopBases ↔ ∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦))))
21ibi 175 . . 3 (𝐵 ∈ TopBases → ∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)))
3 ineq1 3240 . . . . 5 (𝑥 = 𝐶 → (𝑥𝑦) = (𝐶𝑦))
43pweqd 3485 . . . . . . 7 (𝑥 = 𝐶 → 𝒫 (𝑥𝑦) = 𝒫 (𝐶𝑦))
54ineq2d 3247 . . . . . 6 (𝑥 = 𝐶 → (𝐵 ∩ 𝒫 (𝑥𝑦)) = (𝐵 ∩ 𝒫 (𝐶𝑦)))
65unieqd 3717 . . . . 5 (𝑥 = 𝐶 (𝐵 ∩ 𝒫 (𝑥𝑦)) = (𝐵 ∩ 𝒫 (𝐶𝑦)))
73, 6sseq12d 3098 . . . 4 (𝑥 = 𝐶 → ((𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)) ↔ (𝐶𝑦) ⊆ (𝐵 ∩ 𝒫 (𝐶𝑦))))
8 ineq2 3241 . . . . 5 (𝑦 = 𝐷 → (𝐶𝑦) = (𝐶𝐷))
98pweqd 3485 . . . . . . 7 (𝑦 = 𝐷 → 𝒫 (𝐶𝑦) = 𝒫 (𝐶𝐷))
109ineq2d 3247 . . . . . 6 (𝑦 = 𝐷 → (𝐵 ∩ 𝒫 (𝐶𝑦)) = (𝐵 ∩ 𝒫 (𝐶𝐷)))
1110unieqd 3717 . . . . 5 (𝑦 = 𝐷 (𝐵 ∩ 𝒫 (𝐶𝑦)) = (𝐵 ∩ 𝒫 (𝐶𝐷)))
128, 11sseq12d 3098 . . . 4 (𝑦 = 𝐷 → ((𝐶𝑦) ⊆ (𝐵 ∩ 𝒫 (𝐶𝑦)) ↔ (𝐶𝐷) ⊆ (𝐵 ∩ 𝒫 (𝐶𝐷))))
137, 12rspc2v 2776 . . 3 ((𝐶𝐵𝐷𝐵) → (∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)) → (𝐶𝐷) ⊆ (𝐵 ∩ 𝒫 (𝐶𝐷))))
142, 13syl5com 29 . 2 (𝐵 ∈ TopBases → ((𝐶𝐵𝐷𝐵) → (𝐶𝐷) ⊆ (𝐵 ∩ 𝒫 (𝐶𝐷))))
15143impib 1164 1 ((𝐵 ∈ TopBases ∧ 𝐶𝐵𝐷𝐵) → (𝐶𝐷) ⊆ (𝐵 ∩ 𝒫 (𝐶𝐷)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 947   = wceq 1316  wcel 1465  wral 2393  cin 3040  wss 3041  𝒫 cpw 3480   cuni 3706  TopBasesctb 12136
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-in 3047  df-ss 3054  df-pw 3482  df-uni 3707  df-bases 12137
This theorem is referenced by: (None)
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