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Theorem fiinbas 13946
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 3190 . . . . . . . 8 (𝑥𝑦) ⊆ (𝑥𝑦)
2 eleq2 2253 . . . . . . . . . 10 (𝑤 = (𝑥𝑦) → (𝑧𝑤𝑧 ∈ (𝑥𝑦)))
3 sseq1 3193 . . . . . . . . . 10 (𝑤 = (𝑥𝑦) → (𝑤 ⊆ (𝑥𝑦) ↔ (𝑥𝑦) ⊆ (𝑥𝑦)))
42, 3anbi12d 473 . . . . . . . . 9 (𝑤 = (𝑥𝑦) → ((𝑧𝑤𝑤 ⊆ (𝑥𝑦)) ↔ (𝑧 ∈ (𝑥𝑦) ∧ (𝑥𝑦) ⊆ (𝑥𝑦))))
54rspcev 2856 . . . . . . . 8 (((𝑥𝑦) ∈ 𝐵 ∧ (𝑧 ∈ (𝑥𝑦) ∧ (𝑥𝑦) ⊆ (𝑥𝑦))) → ∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦)))
61, 5mpanr2 438 . . . . . . 7 (((𝑥𝑦) ∈ 𝐵𝑧 ∈ (𝑥𝑦)) → ∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦)))
76ralrimiva 2563 . . . . . 6 ((𝑥𝑦) ∈ 𝐵 → ∀𝑧 ∈ (𝑥𝑦)∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦)))
87a1i 9 . . . . 5 (𝐵𝐶 → ((𝑥𝑦) ∈ 𝐵 → ∀𝑧 ∈ (𝑥𝑦)∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦))))
98ralimdv 2558 . . . 4 (𝐵𝐶 → (∀𝑦𝐵 (𝑥𝑦) ∈ 𝐵 → ∀𝑦𝐵𝑧 ∈ (𝑥𝑦)∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦))))
109ralimdv 2558 . . 3 (𝐵𝐶 → (∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ∈ 𝐵 → ∀𝑥𝐵𝑦𝐵𝑧 ∈ (𝑥𝑦)∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦))))
11 isbasis2g 13942 . . 3 (𝐵𝐶 → (𝐵 ∈ TopBases ↔ ∀𝑥𝐵𝑦𝐵𝑧 ∈ (𝑥𝑦)∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦))))
1210, 11sylibrd 169 . 2 (𝐵𝐶 → (∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ∈ 𝐵𝐵 ∈ TopBases))
1312imp 124 1 ((𝐵𝐶 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ∈ 𝐵) → 𝐵 ∈ TopBases)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1364  wcel 2160  wral 2468  wrex 2469  cin 3143  wss 3144  TopBasesctb 13939
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-in 3150  df-ss 3157  df-pw 3592  df-uni 3825  df-bases 13940
This theorem is referenced by:  qtopbasss  14418
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