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Theorem fiinbas 21571
 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 3937 . . . . . . . 8 (𝑥𝑦) ⊆ (𝑥𝑦)
2 eleq2 2878 . . . . . . . . . 10 (𝑤 = (𝑥𝑦) → (𝑧𝑤𝑧 ∈ (𝑥𝑦)))
3 sseq1 3940 . . . . . . . . . 10 (𝑤 = (𝑥𝑦) → (𝑤 ⊆ (𝑥𝑦) ↔ (𝑥𝑦) ⊆ (𝑥𝑦)))
42, 3anbi12d 633 . . . . . . . . 9 (𝑤 = (𝑥𝑦) → ((𝑧𝑤𝑤 ⊆ (𝑥𝑦)) ↔ (𝑧 ∈ (𝑥𝑦) ∧ (𝑥𝑦) ⊆ (𝑥𝑦))))
54rspcev 3571 . . . . . . . 8 (((𝑥𝑦) ∈ 𝐵 ∧ (𝑧 ∈ (𝑥𝑦) ∧ (𝑥𝑦) ⊆ (𝑥𝑦))) → ∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦)))
61, 5mpanr2 703 . . . . . . 7 (((𝑥𝑦) ∈ 𝐵𝑧 ∈ (𝑥𝑦)) → ∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦)))
76ralrimiva 3149 . . . . . 6 ((𝑥𝑦) ∈ 𝐵 → ∀𝑧 ∈ (𝑥𝑦)∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦)))
87a1i 11 . . . . 5 (𝐵𝐶 → ((𝑥𝑦) ∈ 𝐵 → ∀𝑧 ∈ (𝑥𝑦)∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦))))
98ralimdv 3145 . . . 4 (𝐵𝐶 → (∀𝑦𝐵 (𝑥𝑦) ∈ 𝐵 → ∀𝑦𝐵𝑧 ∈ (𝑥𝑦)∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦))))
109ralimdv 3145 . . 3 (𝐵𝐶 → (∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ∈ 𝐵 → ∀𝑥𝐵𝑦𝐵𝑧 ∈ (𝑥𝑦)∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦))))
11 isbasis2g 21567 . . 3 (𝐵𝐶 → (𝐵 ∈ TopBases ↔ ∀𝑥𝐵𝑦𝐵𝑧 ∈ (𝑥𝑦)∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦))))
1210, 11sylibrd 262 . 2 (𝐵𝐶 → (∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ∈ 𝐵𝐵 ∈ TopBases))
1312imp 410 1 ((𝐵𝐶 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ∈ 𝐵) → 𝐵 ∈ TopBases)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107   ∩ cin 3880   ⊆ wss 3881  TopBasesctb 21564 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-in 3888  df-ss 3898  df-pw 4499  df-uni 4802  df-bases 21565 This theorem is referenced by:  fibas  21596  qtopbaslem  23378  ontopbas  33925  isbasisrelowl  34815
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