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

Theorem cfilufbas 22901
Description: A Cauchy filter base is a filter base. (Contributed by Thierry Arnoux, 19-Nov-2017.)
Assertion
Ref Expression
cfilufbas ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu𝑈)) → 𝐹 ∈ (fBas‘𝑋))

Proof of Theorem cfilufbas
Dummy variables 𝑣 𝑎 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iscfilu 22900 . 2 (𝑈 ∈ (UnifOn‘𝑋) → (𝐹 ∈ (CauFilu𝑈) ↔ (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑣𝑈𝑎𝐹 (𝑎 × 𝑎) ⊆ 𝑣)))
21simprbda 502 1 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐹 ∈ (CauFilu𝑈)) → 𝐹 ∈ (fBas‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wcel 2115  wral 3133  wrex 3134  wss 3920   × cxp 5541  cfv 6344  fBascfbas 20536  UnifOncust 22811  CauFiluccfilu 22898
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7456
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3483  df-sbc 3760  df-dif 3923  df-un 3925  df-in 3927  df-ss 3937  df-nul 4278  df-if 4452  df-pw 4525  df-sn 4552  df-pr 4554  df-op 4558  df-uni 4826  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-iota 6303  df-fun 6346  df-fn 6347  df-fv 6352  df-ust 22812  df-cfilu 22899
This theorem is referenced by:  fmucnd  22904  cfilufg  22905  cfilucfil  23172
  Copyright terms: Public domain W3C validator