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Theorem basis1 12289
 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 12286 . . . 4 (𝐵 ∈ TopBases → (𝐵 ∈ TopBases ↔ ∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦))))
21ibi 175 . . 3 (𝐵 ∈ TopBases → ∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)))
3 ineq1 3277 . . . . 5 (𝑥 = 𝐶 → (𝑥𝑦) = (𝐶𝑦))
43pweqd 3522 . . . . . . 7 (𝑥 = 𝐶 → 𝒫 (𝑥𝑦) = 𝒫 (𝐶𝑦))
54ineq2d 3284 . . . . . 6 (𝑥 = 𝐶 → (𝐵 ∩ 𝒫 (𝑥𝑦)) = (𝐵 ∩ 𝒫 (𝐶𝑦)))
65unieqd 3757 . . . . 5 (𝑥 = 𝐶 (𝐵 ∩ 𝒫 (𝑥𝑦)) = (𝐵 ∩ 𝒫 (𝐶𝑦)))
73, 6sseq12d 3135 . . . 4 (𝑥 = 𝐶 → ((𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)) ↔ (𝐶𝑦) ⊆ (𝐵 ∩ 𝒫 (𝐶𝑦))))
8 ineq2 3278 . . . . 5 (𝑦 = 𝐷 → (𝐶𝑦) = (𝐶𝐷))
98pweqd 3522 . . . . . . 7 (𝑦 = 𝐷 → 𝒫 (𝐶𝑦) = 𝒫 (𝐶𝐷))
109ineq2d 3284 . . . . . 6 (𝑦 = 𝐷 → (𝐵 ∩ 𝒫 (𝐶𝑦)) = (𝐵 ∩ 𝒫 (𝐶𝐷)))
1110unieqd 3757 . . . . 5 (𝑦 = 𝐷 (𝐵 ∩ 𝒫 (𝐶𝑦)) = (𝐵 ∩ 𝒫 (𝐶𝐷)))
128, 11sseq12d 3135 . . . 4 (𝑦 = 𝐷 → ((𝐶𝑦) ⊆ (𝐵 ∩ 𝒫 (𝐶𝑦)) ↔ (𝐶𝐷) ⊆ (𝐵 ∩ 𝒫 (𝐶𝐷))))
137, 12rspc2v 2808 . . 3 ((𝐶𝐵𝐷𝐵) → (∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ⊆ (𝐵 ∩ 𝒫 (𝑥𝑦)) → (𝐶𝐷) ⊆ (𝐵 ∩ 𝒫 (𝐶𝐷))))
142, 13syl5com 29 . 2 (𝐵 ∈ TopBases → ((𝐶𝐵𝐷𝐵) → (𝐶𝐷) ⊆ (𝐵 ∩ 𝒫 (𝐶𝐷))))
15143impib 1180 1 ((𝐵 ∈ TopBases ∧ 𝐶𝐵𝐷𝐵) → (𝐶𝐷) ⊆ (𝐵 ∩ 𝒫 (𝐶𝐷)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ∧ w3a 963   = wceq 1332   ∈ wcel 1481  ∀wral 2418   ∩ cin 3077   ⊆ wss 3078  𝒫 cpw 3517  ∪ cuni 3746  TopBasesctb 12284 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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2123 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1738  df-clab 2128  df-cleq 2134  df-clel 2137  df-nfc 2272  df-ral 2423  df-rex 2424  df-v 2693  df-in 3084  df-ss 3091  df-pw 3519  df-uni 3747  df-bases 12285 This theorem is referenced by: (None)
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