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Theorem cuspcvg 21915
 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 simpr 476 . . . . 5 ((𝑊 ∈ CUnifSp ∧ 𝐶 ∈ (Fil‘𝐵)) → 𝐶 ∈ (Fil‘𝐵))
2 cuspcvg.1 . . . . . 6 𝐵 = (Base‘𝑊)
32fveq2i 6106 . . . . 5 (Fil‘𝐵) = (Fil‘(Base‘𝑊))
41, 3syl6eleq 2698 . . . 4 ((𝑊 ∈ CUnifSp ∧ 𝐶 ∈ (Fil‘𝐵)) → 𝐶 ∈ (Fil‘(Base‘𝑊)))
5 iscusp 21913 . . . . . 6 (𝑊 ∈ CUnifSp ↔ (𝑊 ∈ UnifSp ∧ ∀𝑐 ∈ (Fil‘(Base‘𝑊))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅)))
65simprbi 479 . . . . 5 (𝑊 ∈ CUnifSp → ∀𝑐 ∈ (Fil‘(Base‘𝑊))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅))
76adantr 480 . . . 4 ((𝑊 ∈ CUnifSp ∧ 𝐶 ∈ (Fil‘𝐵)) → ∀𝑐 ∈ (Fil‘(Base‘𝑊))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅))
8 eleq1 2676 . . . . . 6 (𝑐 = 𝐶 → (𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) ↔ 𝐶 ∈ (CauFilu‘(UnifSt‘𝑊))))
9 cuspcvg.2 . . . . . . . . . 10 𝐽 = (TopOpen‘𝑊)
109eqcomi 2619 . . . . . . . . 9 (TopOpen‘𝑊) = 𝐽
1110a1i 11 . . . . . . . 8 (𝑐 = 𝐶 → (TopOpen‘𝑊) = 𝐽)
12 id 22 . . . . . . . 8 (𝑐 = 𝐶𝑐 = 𝐶)
1311, 12oveq12d 6567 . . . . . . 7 (𝑐 = 𝐶 → ((TopOpen‘𝑊) fLim 𝑐) = (𝐽 fLim 𝐶))
1413neeq1d 2841 . . . . . 6 (𝑐 = 𝐶 → (((TopOpen‘𝑊) fLim 𝑐) ≠ ∅ ↔ (𝐽 fLim 𝐶) ≠ ∅))
158, 14imbi12d 333 . . . . 5 (𝑐 = 𝐶 → ((𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅) ↔ (𝐶 ∈ (CauFilu‘(UnifSt‘𝑊)) → (𝐽 fLim 𝐶) ≠ ∅)))
1615rspcva 3280 . . . 4 ((𝐶 ∈ (Fil‘(Base‘𝑊)) ∧ ∀𝑐 ∈ (Fil‘(Base‘𝑊))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅)) → (𝐶 ∈ (CauFilu‘(UnifSt‘𝑊)) → (𝐽 fLim 𝐶) ≠ ∅))
174, 7, 16syl2anc 691 . . 3 ((𝑊 ∈ CUnifSp ∧ 𝐶 ∈ (Fil‘𝐵)) → (𝐶 ∈ (CauFilu‘(UnifSt‘𝑊)) → (𝐽 fLim 𝐶) ≠ ∅))
18173impia 1253 . 2 ((𝑊 ∈ CUnifSp ∧ 𝐶 ∈ (Fil‘𝐵) ∧ 𝐶 ∈ (CauFilu‘(UnifSt‘𝑊))) → (𝐽 fLim 𝐶) ≠ ∅)
19183com23 1263 1 ((𝑊 ∈ CUnifSp ∧ 𝐶 ∈ (CauFilu‘(UnifSt‘𝑊)) ∧ 𝐶 ∈ (Fil‘𝐵)) → (𝐽 fLim 𝐶) ≠ ∅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∅c0 3874  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  TopOpenctopn 15905  Filcfil 21459   fLim cflim 21548  UnifStcuss 21867  UnifSpcusp 21868  CauFiluccfilu 21900  CUnifSpccusp 21911 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-cusp 21912 This theorem is referenced by:  cnextucn  21917
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