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

Definition df-xms 22859
Description: Define the (proper) class of extended metric spaces. (Contributed by Mario Carneiro, 2-Sep-2015.)
Assertion
Ref Expression
df-xms ∞MetSp = {𝑓 ∈ TopSp ∣ (TopOpen‘𝑓) = (MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))))}

Detailed syntax breakdown of Definition df-xms
StepHypRef Expression
1 cxms 22856 . 2 class ∞MetSp
2 vf . . . . . 6 setvar 𝑓
32cv 1527 . . . . 5 class 𝑓
4 ctopn 16685 . . . . 5 class TopOpen
53, 4cfv 6349 . . . 4 class (TopOpen‘𝑓)
6 cds 16564 . . . . . . 7 class dist
73, 6cfv 6349 . . . . . 6 class (dist‘𝑓)
8 cbs 16473 . . . . . . . 8 class Base
93, 8cfv 6349 . . . . . . 7 class (Base‘𝑓)
109, 9cxp 5547 . . . . . 6 class ((Base‘𝑓) × (Base‘𝑓))
117, 10cres 5551 . . . . 5 class ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓)))
12 cmopn 20465 . . . . 5 class MetOpen
1311, 12cfv 6349 . . . 4 class (MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))))
145, 13wceq 1528 . . 3 wff (TopOpen‘𝑓) = (MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))))
15 ctps 21470 . . 3 class TopSp
1614, 2, 15crab 3142 . 2 class {𝑓 ∈ TopSp ∣ (TopOpen‘𝑓) = (MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))))}
171, 16wceq 1528 1 wff ∞MetSp = {𝑓 ∈ TopSp ∣ (TopOpen‘𝑓) = (MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))))}
Colors of variables: wff setvar class
This definition is referenced by:  isxms  22986
  Copyright terms: Public domain W3C validator