MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cfiluexsm Structured version   Visualization version   GIF version
Theorem cfiluexsm 24116
Description: For a Cauchy filter base and any entourage 𝑉, there is an element of the filter small in 𝑉. (Contributed by Thierry Arnoux, 19-Nov-2017.)
Assertion
Ref Expression
cfiluexsm ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝐹 ∈ (CauFiluβ€˜π‘ˆ) ∧ 𝑉 ∈ π‘ˆ) β†’ βˆƒπ‘Ž ∈ 𝐹 (π‘Ž Γ— π‘Ž) βŠ† 𝑉)
Distinct variable groups:   𝐹,π‘Ž   𝑉,π‘Ž
Allowed substitution hints:   π‘ˆ(π‘Ž)   𝑋(π‘Ž)
Proof of Theorem cfiluexsm
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 iscfilu 24114 . . . 4 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ (𝐹 ∈ (CauFiluβ€˜π‘ˆ) ↔ (𝐹 ∈ (fBasβ€˜π‘‹) ∧ βˆ€π‘£ ∈ π‘ˆ βˆƒπ‘Ž ∈ 𝐹 (π‘Ž Γ— π‘Ž) βŠ† 𝑣)))
21simplbda 499 . . 3 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝐹 ∈ (CauFiluβ€˜π‘ˆ)) β†’ βˆ€π‘£ ∈ π‘ˆ βˆƒπ‘Ž ∈ 𝐹 (π‘Ž Γ— π‘Ž) βŠ† 𝑣)
323adant3 1129 . 2 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝐹 ∈ (CauFiluβ€˜π‘ˆ) ∧ 𝑉 ∈ π‘ˆ) β†’ βˆ€π‘£ ∈ π‘ˆ βˆƒπ‘Ž ∈ 𝐹 (π‘Ž Γ— π‘Ž) βŠ† 𝑣)
4 sseq2 4000 . . . . 5 (𝑣 = 𝑉 β†’ ((π‘Ž Γ— π‘Ž) βŠ† 𝑣 ↔ (π‘Ž Γ— π‘Ž) βŠ† 𝑉))
54rexbidv 3170 . . . 4 (𝑣 = 𝑉 β†’ (βˆƒπ‘Ž ∈ 𝐹 (π‘Ž Γ— π‘Ž) βŠ† 𝑣 ↔ βˆƒπ‘Ž ∈ 𝐹 (π‘Ž Γ— π‘Ž) βŠ† 𝑉))
65rspcv 3600 . . 3 (𝑉 ∈ π‘ˆ β†’ (βˆ€π‘£ ∈ π‘ˆ βˆƒπ‘Ž ∈ 𝐹 (π‘Ž Γ— π‘Ž) βŠ† 𝑣 β†’ βˆƒπ‘Ž ∈ 𝐹 (π‘Ž Γ— π‘Ž) βŠ† 𝑉))
763ad2ant3 1132 . 2 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝐹 ∈ (CauFiluβ€˜π‘ˆ) ∧ 𝑉 ∈ π‘ˆ) β†’ (βˆ€π‘£ ∈ π‘ˆ βˆƒπ‘Ž ∈ 𝐹 (π‘Ž Γ— π‘Ž) βŠ† 𝑣 β†’ βˆƒπ‘Ž ∈ 𝐹 (π‘Ž Γ— π‘Ž) βŠ† 𝑉))
83, 7mpd 15 1 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝐹 ∈ (CauFiluβ€˜π‘ˆ) ∧ 𝑉 ∈ π‘ˆ) β†’ βˆƒπ‘Ž ∈ 𝐹 (π‘Ž Γ— π‘Ž) βŠ† 𝑉)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3053  βˆƒwrex 3062   βŠ† wss 3940   Γ— cxp 5664  β€˜cfv 6533  fBascfbas 21215  UnifOncust 24025  CauFiluccfilu 24112
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-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-iota 6485  df-fun 6535  df-fv 6541  df-ust 24026  df-cfilu 24113
This theorem is referenced by:  fmucnd  24118  cfilucfil  24389
  Copyright terms: Public domain W3C validator