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Theorem iscusp2 23199
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 23196 . 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 6720 . . . 4 (Fil‘𝐵) = (Fil‘(Base‘𝑊))
4 iscusp2.2 . . . . . . 7 𝑈 = (UnifSt‘𝑊)
54fveq2i 6720 . . . . . 6 (CauFilu𝑈) = (CauFilu‘(UnifSt‘𝑊))
65eleq2i 2829 . . . . 5 (𝑐 ∈ (CauFilu𝑈) ↔ 𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)))
7 iscusp2.3 . . . . . . 7 𝐽 = (TopOpen‘𝑊)
87oveq1i 7223 . . . . . 6 (𝐽 fLim 𝑐) = ((TopOpen‘𝑊) fLim 𝑐)
98neeq1i 3005 . . . . 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 3156 . . 3 (∀𝑐 ∈ (Fil‘𝐵)(𝑐 ∈ (CauFilu𝑈) → (𝐽 fLim 𝑐) ≠ ∅) ↔ ∀𝑐 ∈ (Fil‘(Base‘𝑊))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑊)) → ((TopOpen‘𝑊) fLim 𝑐) ≠ ∅))
1211anbi2i 626 . 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 1543  wcel 2110  wne 2940  wral 3061  c0 4237  cfv 6380  (class class class)co 7213  Basecbs 16760  TopOpenctopn 16926  Filcfil 22742   fLim cflim 22831  UnifStcuss 23151  UnifSpcusp 23152  CauFiluccfilu 23183  CUnifSpccusp 23194
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3066  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-iota 6338  df-fv 6388  df-ov 7216  df-cusp 23195
This theorem is referenced by:  cmetcusp1  24250
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