MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fiinbas Structured version   Visualization version   GIF version

Theorem fiinbas 21050
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 3785 . . . . . . . 8 (𝑥𝑦) ⊆ (𝑥𝑦)
2 eleq2 2833 . . . . . . . . . 10 (𝑤 = (𝑥𝑦) → (𝑧𝑤𝑧 ∈ (𝑥𝑦)))
3 sseq1 3788 . . . . . . . . . 10 (𝑤 = (𝑥𝑦) → (𝑤 ⊆ (𝑥𝑦) ↔ (𝑥𝑦) ⊆ (𝑥𝑦)))
42, 3anbi12d 624 . . . . . . . . 9 (𝑤 = (𝑥𝑦) → ((𝑧𝑤𝑤 ⊆ (𝑥𝑦)) ↔ (𝑧 ∈ (𝑥𝑦) ∧ (𝑥𝑦) ⊆ (𝑥𝑦))))
54rspcev 3462 . . . . . . . 8 (((𝑥𝑦) ∈ 𝐵 ∧ (𝑧 ∈ (𝑥𝑦) ∧ (𝑥𝑦) ⊆ (𝑥𝑦))) → ∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦)))
61, 5mpanr2 695 . . . . . . 7 (((𝑥𝑦) ∈ 𝐵𝑧 ∈ (𝑥𝑦)) → ∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦)))
76ralrimiva 3113 . . . . . 6 ((𝑥𝑦) ∈ 𝐵 → ∀𝑧 ∈ (𝑥𝑦)∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦)))
87a1i 11 . . . . 5 (𝐵𝐶 → ((𝑥𝑦) ∈ 𝐵 → ∀𝑧 ∈ (𝑥𝑦)∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦))))
98ralimdv 3110 . . . 4 (𝐵𝐶 → (∀𝑦𝐵 (𝑥𝑦) ∈ 𝐵 → ∀𝑦𝐵𝑧 ∈ (𝑥𝑦)∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦))))
109ralimdv 3110 . . 3 (𝐵𝐶 → (∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ∈ 𝐵 → ∀𝑥𝐵𝑦𝐵𝑧 ∈ (𝑥𝑦)∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦))))
11 isbasis2g 21046 . . 3 (𝐵𝐶 → (𝐵 ∈ TopBases ↔ ∀𝑥𝐵𝑦𝐵𝑧 ∈ (𝑥𝑦)∃𝑤𝐵 (𝑧𝑤𝑤 ⊆ (𝑥𝑦))))
1210, 11sylibrd 250 . 2 (𝐵𝐶 → (∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ∈ 𝐵𝐵 ∈ TopBases))
1312imp 395 1 ((𝐵𝐶 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥𝑦) ∈ 𝐵) → 𝐵 ∈ TopBases)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1652  wcel 2155  wral 3055  wrex 3056  cin 3733  wss 3734  TopBasesctb 21043
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-v 3352  df-in 3741  df-ss 3748  df-pw 4319  df-uni 4597  df-bases 21044
This theorem is referenced by:  fibas  21075  qtopbaslem  22855  ontopbas  32887  isbasisrelowl  33660
  Copyright terms: Public domain W3C validator