MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-cusp Structured version   Visualization version   GIF version

Definition df-cusp 22149
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 22148 . 2 class CUnifSp
2 vc . . . . . . 7 setvar 𝑐
32cv 1522 . . . . . 6 class 𝑐
4 vw . . . . . . . . 9 setvar 𝑤
54cv 1522 . . . . . . . 8 class 𝑤
6 cuss 22104 . . . . . . . 8 class UnifSt
75, 6cfv 5926 . . . . . . 7 class (UnifSt‘𝑤)
8 ccfilu 22137 . . . . . . 7 class CauFilu
97, 8cfv 5926 . . . . . 6 class (CauFilu‘(UnifSt‘𝑤))
103, 9wcel 2030 . . . . 5 wff 𝑐 ∈ (CauFilu‘(UnifSt‘𝑤))
11 ctopn 16129 . . . . . . . 8 class TopOpen
125, 11cfv 5926 . . . . . . 7 class (TopOpen‘𝑤)
13 cflim 21785 . . . . . . 7 class fLim
1412, 3, 13co 6690 . . . . . 6 class ((TopOpen‘𝑤) fLim 𝑐)
15 c0 3948 . . . . . 6 class
1614, 15wne 2823 . . . . 5 wff ((TopOpen‘𝑤) fLim 𝑐) ≠ ∅
1710, 16wi 4 . . . 4 wff (𝑐 ∈ (CauFilu‘(UnifSt‘𝑤)) → ((TopOpen‘𝑤) fLim 𝑐) ≠ ∅)
18 cbs 15904 . . . . . 6 class Base
195, 18cfv 5926 . . . . 5 class (Base‘𝑤)
20 cfil 21696 . . . . 5 class Fil
2119, 20cfv 5926 . . . 4 class (Fil‘(Base‘𝑤))
2217, 2, 21wral 2941 . . 3 wff 𝑐 ∈ (Fil‘(Base‘𝑤))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑤)) → ((TopOpen‘𝑤) fLim 𝑐) ≠ ∅)
23 cusp 22105 . . 3 class UnifSp
2422, 4, 23crab 2945 . 2 class {𝑤 ∈ UnifSp ∣ ∀𝑐 ∈ (Fil‘(Base‘𝑤))(𝑐 ∈ (CauFilu‘(UnifSt‘𝑤)) → ((TopOpen‘𝑤) fLim 𝑐) ≠ ∅)}
251, 24wceq 1523 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  22150
  Copyright terms: Public domain W3C validator