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Theorem iscusp2 21916
 Description: The predicate "𝑊 is a complete uniform space." (Contributed by Thierry Arnoux, 15-Dec-2017.)
Hypotheses
Ref Expression
iscusp2.1 𝐵 = (Base‘𝑊)
iscusp2.2 𝑈 = (UnifSt‘𝑊)
iscusp2.3 𝐽 = (TopOpen‘𝑊)
Assertion
Ref Expression
iscusp2 (𝑊 ∈ CUnifSp ↔ (𝑊 ∈ UnifSp ∧ ∀𝑐 ∈ (Fil‘𝐵)(𝑐 ∈ (CauFilu𝑈) → (𝐽 fLim 𝑐) ≠ ∅)))
Distinct variable group:   𝑊,𝑐
Allowed substitution hints:   𝐵(𝑐)   𝑈(𝑐)   𝐽(𝑐)

Proof of Theorem iscusp2
StepHypRef Expression
1 iscusp 21913 . 2 (𝑊 ∈ CUnifSp ↔ (𝑊 ∈ UnifSp ∧ ∀𝑐 ∈ (Fil‘(Base‘𝑊))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅)))
2 iscusp2.1 . . . . 5 𝐵 = (Base‘𝑊)
32fveq2i 6106 . . . 4 (Fil‘𝐵) = (Fil‘(Base‘𝑊))
4 iscusp2.2 . . . . . . 7 𝑈 = (UnifSt‘𝑊)
54fveq2i 6106 . . . . . 6 (CauFilu𝑈) = (CauFilu‘(UnifSt‘𝑊))
65eleq2i 2680 . . . . 5 (𝑐 ∈ (CauFilu𝑈) ↔ 𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)))
7 iscusp2.3 . . . . . . 7 𝐽 = (TopOpen‘𝑊)
87oveq1i 6559 . . . . . 6 (𝐽 fLim 𝑐) = ((TopOpen‘𝑊) fLim 𝑐)
98neeq1i 2846 . . . . 5 ((𝐽 fLim 𝑐) ≠ ∅ ↔ ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅)
106, 9imbi12i 339 . . . 4 ((𝑐 ∈ (CauFilu𝑈) → (𝐽 fLim 𝑐) ≠ ∅) ↔ (𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅))
113, 10raleqbii 2973 . . 3 (∀𝑐 ∈ (Fil‘𝐵)(𝑐 ∈ (CauFilu𝑈) → (𝐽 fLim 𝑐) ≠ ∅) ↔ ∀𝑐 ∈ (Fil‘(Base‘𝑊))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅))
1211anbi2i 726 . 2 ((𝑊 ∈ UnifSp ∧ ∀𝑐 ∈ (Fil‘𝐵)(𝑐 ∈ (CauFilu𝑈) → (𝐽 fLim 𝑐) ≠ ∅)) ↔ (𝑊 ∈ UnifSp ∧ ∀𝑐 ∈ (Fil‘(Base‘𝑊))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅)))
131, 12bitr4i 266 1 (𝑊 ∈ CUnifSp ↔ (𝑊 ∈ UnifSp ∧ ∀𝑐 ∈ (Fil‘𝐵)(𝑐 ∈ (CauFilu𝑈) → (𝐽 fLim 𝑐) ≠ ∅)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = 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:  cmetcusp1  22957
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