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Theorem iscusp2 22949
 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 22946 . 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 6658 . . . 4 (Fil‘𝐵) = (Fil‘(Base‘𝑊))
4 iscusp2.2 . . . . . . 7 𝑈 = (UnifSt‘𝑊)
54fveq2i 6658 . . . . . 6 (CauFilu𝑈) = (CauFilu‘(UnifSt‘𝑊))
65eleq2i 2881 . . . . 5 (𝑐 ∈ (CauFilu𝑈) ↔ 𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)))
7 iscusp2.3 . . . . . . 7 𝐽 = (TopOpen‘𝑊)
87oveq1i 7155 . . . . . 6 (𝐽 fLim 𝑐) = ((TopOpen‘𝑊) fLim 𝑐)
98neeq1i 3051 . . . . 5 ((𝐽 fLim 𝑐) ≠ ∅ ↔ ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅)
106, 9imbi12i 354 . . . 4 ((𝑐 ∈ (CauFilu𝑈) → (𝐽 fLim 𝑐) ≠ ∅) ↔ (𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅))
113, 10raleqbii 3197 . . 3 (∀𝑐 ∈ (Fil‘𝐵)(𝑐 ∈ (CauFilu𝑈) → (𝐽 fLim 𝑐) ≠ ∅) ↔ ∀𝑐 ∈ (Fil‘(Base‘𝑊))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅))
1211anbi2i 625 . 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 281 1 (𝑊 ∈ CUnifSp ↔ (𝑊 ∈ UnifSp ∧ ∀𝑐 ∈ (Fil‘𝐵)(𝑐 ∈ (CauFilu𝑈) → (𝐽 fLim 𝑐) ≠ ∅)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  ∅c0 4246  ‘cfv 6332  (class class class)co 7145  Basecbs 16495  TopOpenctopn 16707  Filcfil 22491   fLim cflim 22580  UnifStcuss 22900  UnifSpcusp 22901  CauFiluccfilu 22933  CUnifSpccusp 22944 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 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 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rab 3115  df-v 3444  df-un 3888  df-in 3890  df-ss 3900  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-br 5035  df-iota 6291  df-fv 6340  df-ov 7148  df-cusp 22945 This theorem is referenced by:  cmetcusp1  23998
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