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Theorem cuspcvg 22837
Description: In a complete uniform space, any Cauchy filter 𝐶 has a limit. (Contributed by Thierry Arnoux, 3-Dec-2017.)
Hypotheses
Ref Expression
cuspcvg.1 𝐵 = (Base‘𝑊)
cuspcvg.2 𝐽 = (TopOpen‘𝑊)
Assertion
Ref Expression
cuspcvg ((𝑊 ∈ CUnifSp ∧ 𝐶 ∈ (CauFilu‘(UnifSt‘𝑊)) ∧ 𝐶 ∈ (Fil‘𝐵)) → (𝐽 fLim 𝐶) ≠ ∅)

Proof of Theorem cuspcvg
Dummy variable 𝑐 is distinct from all other variables.
StepHypRef Expression
1 eleq1 2897 . . . . 5 (𝑐 = 𝐶 → (𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) ↔ 𝐶 ∈ (CauFilu‘(UnifSt‘𝑊))))
2 cuspcvg.2 . . . . . . . . 9 𝐽 = (TopOpen‘𝑊)
32eqcomi 2827 . . . . . . . 8 (TopOpen‘𝑊) = 𝐽
43a1i 11 . . . . . . 7 (𝑐 = 𝐶 → (TopOpen‘𝑊) = 𝐽)
5 id 22 . . . . . . 7 (𝑐 = 𝐶𝑐 = 𝐶)
64, 5oveq12d 7163 . . . . . 6 (𝑐 = 𝐶 → ((TopOpen‘𝑊) fLim 𝑐) = (𝐽 fLim 𝐶))
76neeq1d 3072 . . . . 5 (𝑐 = 𝐶 → (((TopOpen‘𝑊) fLim 𝑐) ≠ ∅ ↔ (𝐽 fLim 𝐶) ≠ ∅))
81, 7imbi12d 346 . . . 4 (𝑐 = 𝐶 → ((𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅) ↔ (𝐶 ∈ (CauFilu‘(UnifSt‘𝑊)) → (𝐽 fLim 𝐶) ≠ ∅)))
9 iscusp 22835 . . . . . 6 (𝑊 ∈ CUnifSp ↔ (𝑊 ∈ UnifSp ∧ ∀𝑐 ∈ (Fil‘(Base‘𝑊))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅)))
109simprbi 497 . . . . 5 (𝑊 ∈ CUnifSp → ∀𝑐 ∈ (Fil‘(Base‘𝑊))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅))
1110adantr 481 . . . 4 ((𝑊 ∈ CUnifSp ∧ 𝐶 ∈ (Fil‘𝐵)) → ∀𝑐 ∈ (Fil‘(Base‘𝑊))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅))
12 simpr 485 . . . . 5 ((𝑊 ∈ CUnifSp ∧ 𝐶 ∈ (Fil‘𝐵)) → 𝐶 ∈ (Fil‘𝐵))
13 cuspcvg.1 . . . . . 6 𝐵 = (Base‘𝑊)
1413fveq2i 6666 . . . . 5 (Fil‘𝐵) = (Fil‘(Base‘𝑊))
1512, 14eleqtrdi 2920 . . . 4 ((𝑊 ∈ CUnifSp ∧ 𝐶 ∈ (Fil‘𝐵)) → 𝐶 ∈ (Fil‘(Base‘𝑊)))
168, 11, 15rspcdva 3622 . . 3 ((𝑊 ∈ CUnifSp ∧ 𝐶 ∈ (Fil‘𝐵)) → (𝐶 ∈ (CauFilu‘(UnifSt‘𝑊)) → (𝐽 fLim 𝐶) ≠ ∅))
17163impia 1109 . 2 ((𝑊 ∈ CUnifSp ∧ 𝐶 ∈ (Fil‘𝐵) ∧ 𝐶 ∈ (CauFilu‘(UnifSt‘𝑊))) → (𝐽 fLim 𝐶) ≠ ∅)
18173com23 1118 1 ((𝑊 ∈ CUnifSp ∧ 𝐶 ∈ (CauFilu‘(UnifSt‘𝑊)) ∧ 𝐶 ∈ (Fil‘𝐵)) → (𝐽 fLim 𝐶) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105  wne 3013  wral 3135  c0 4288  cfv 6348  (class class class)co 7145  Basecbs 16471  TopOpenctopn 16683  Filcfil 22381   fLim cflim 22470  UnifStcuss 22789  UnifSpcusp 22790  CauFiluccfilu 22822  CUnifSpccusp 22833
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356  df-ov 7148  df-cusp 22834
This theorem is referenced by:  cnextucn  22839
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