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Theorem fiinbas 22875
Description: If a set is closed under finite intersection, then it is a basis for a topology. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
fiinbas ((𝐵𝐶 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ∈ 𝐵) → 𝐵 ∈ TopBases)
Distinct variable groups:   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦

Proof of Theorem fiinbas
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssid 4004 . . . . . . . 8 (𝑥𝑦) ⊆ (𝑥𝑦)
2 eleq2 2818 . . . . . . . . . 10 (𝑤 = (𝑥𝑦) → (𝑧𝑤𝑧 ∈ (𝑥𝑦)))
3 sseq1 4007 . . . . . . . . . 10 (𝑤 = (𝑥𝑦) → (𝑤 ⊆ (𝑥𝑦) ↔ (𝑥𝑦) ⊆ (𝑥𝑦)))
42, 3anbi12d 630 . . . . . . . . 9 (𝑤 = (𝑥𝑦) → ((𝑧𝑤𝑤 ⊆ (𝑥𝑦)) ↔ (𝑧 ∈ (𝑥𝑦) ∧ (𝑥𝑦) ⊆ (𝑥𝑦))))
54rspcev 3611 . . . . . . . 8 (((𝑥𝑦) ∈ 𝐵 ∧ (𝑧 ∈ (𝑥𝑦) ∧ (𝑥𝑦) ⊆ (𝑥𝑦))) → ∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦)))
61, 5mpanr2 702 . . . . . . 7 (((𝑥𝑦) ∈ 𝐵𝑧 ∈ (𝑥𝑦)) → ∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦)))
76ralrimiva 3143 . . . . . 6 ((𝑥𝑦) ∈ 𝐵 → ∀𝑧 ∈ (𝑥𝑦)∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦)))
87a1i 11 . . . . 5 (𝐵𝐶 → ((𝑥𝑦) ∈ 𝐵 → ∀𝑧 ∈ (𝑥𝑦)∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦))))
98ralimdv 3166 . . . 4 (𝐵𝐶 → (∀𝑦𝐵 (𝑥𝑦) ∈ 𝐵 → ∀𝑦𝐵𝑧 ∈ (𝑥𝑦)∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦))))
109ralimdv 3166 . . 3 (𝐵𝐶 → (∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ∈ 𝐵 → ∀𝑥𝐵𝑦𝐵𝑧 ∈ (𝑥𝑦)∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦))))
11 isbasis2g 22871 . . 3 (𝐵𝐶 → (𝐵 ∈ TopBases ↔ ∀𝑥𝐵𝑦𝐵𝑧 ∈ (𝑥𝑦)∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦))))
1210, 11sylibrd 258 . 2 (𝐵𝐶 → (∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ∈ 𝐵𝐵 ∈ TopBases))
1312imp 405 1 ((𝐵𝐶 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ∈ 𝐵) → 𝐵 ∈ TopBases)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  wral 3058  wrex 3067  cin 3948  wss 3949  TopBasesctb 22868
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-in 3956  df-ss 3966  df-pw 4608  df-uni 4913  df-bases 22869
This theorem is referenced by:  fibas  22900  qtopbaslem  24695  ontopbas  35945  isbasisrelowl  36870
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