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Definition df-cusp 21854
Description: Define the class of all complete uniform spaces. Definition 3 of [BourbakiTop1] p. II.15. (Contributed by Thierry Arnoux, 1-Dec-2017.)
Assertion
Ref Expression
df-cusp CUnifSp = {𝑤 ∈ UnifSp ∣ ∀𝑐 ∈ (Fil‘(Base‘𝑤))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑤)) → ((TopOpen‘𝑤) fLim 𝑐) ≠ ∅)}
Distinct variable group:   𝑤,𝑐

Detailed syntax breakdown of Definition df-cusp
StepHypRef Expression
1 ccusp 21853 . 2 class CUnifSp
2 vc . . . . . . 7 setvar 𝑐
32cv 1473 . . . . . 6 class 𝑐
4 vw . . . . . . . . 9 setvar 𝑤
54cv 1473 . . . . . . . 8 class 𝑤
6 cuss 21809 . . . . . . . 8 class UnifSt
75, 6cfv 5790 . . . . . . 7 class (UnifSt‘𝑤)
8 ccfilu 21842 . . . . . . 7 class CauFilu
97, 8cfv 5790 . . . . . 6 class (CauFilu‘(UnifSt‘𝑤))
103, 9wcel 1976 . . . . 5 wff 𝑐 ∈ (CauFilu‘(UnifSt‘𝑤))
11 ctopn 15851 . . . . . . . 8 class TopOpen
125, 11cfv 5790 . . . . . . 7 class (TopOpen‘𝑤)
13 cflim 21490 . . . . . . 7 class fLim
1412, 3, 13co 6527 . . . . . 6 class ((TopOpen‘𝑤) fLim 𝑐)
15 c0 3873 . . . . . 6 class
1614, 15wne 2779 . . . . 5 wff ((TopOpen‘𝑤) fLim 𝑐) ≠ ∅
1710, 16wi 4 . . . 4 wff (𝑐 ∈ (CauFilu‘(UnifSt‘𝑤)) → ((TopOpen‘𝑤) fLim 𝑐) ≠ ∅)
18 cbs 15641 . . . . . 6 class Base
195, 18cfv 5790 . . . . 5 class (Base‘𝑤)
20 cfil 21401 . . . . 5 class Fil
2119, 20cfv 5790 . . . 4 class (Fil‘(Base‘𝑤))
2217, 2, 21wral 2895 . . 3 wff 𝑐 ∈ (Fil‘(Base‘𝑤))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑤)) → ((TopOpen‘𝑤) fLim 𝑐) ≠ ∅)
23 cusp 21810 . . 3 class UnifSp
2422, 4, 23crab 2899 . 2 class {𝑤 ∈ UnifSp ∣ ∀𝑐 ∈ (Fil‘(Base‘𝑤))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑤)) → ((TopOpen‘𝑤) fLim 𝑐) ≠ ∅)}
251, 24wceq 1474 1 wff CUnifSp = {𝑤 ∈ UnifSp ∣ ∀𝑐 ∈ (Fil‘(Base‘𝑤))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑤)) → ((TopOpen‘𝑤) fLim 𝑐) ≠ ∅)}
Colors of variables: wff setvar class
This definition is referenced by:  iscusp  21855
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