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Definition df-xms 12545
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 12542 . 2 class ∞MetSp
2 vf . . . . . 6 setvar 𝑓
32cv 1331 . . . . 5 class 𝑓
4 ctopn 12158 . . . . 5 class TopOpen
53, 4cfv 5130 . . . 4 class (TopOpen‘𝑓)
6 cds 12067 . . . . . . 7 class dist
73, 6cfv 5130 . . . . . 6 class (dist‘𝑓)
8 cbs 11996 . . . . . . . 8 class Base
93, 8cfv 5130 . . . . . . 7 class (Base‘𝑓)
109, 9cxp 4544 . . . . . 6 class ((Base‘𝑓) × (Base‘𝑓))
117, 10cres 4548 . . . . 5 class ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓)))
12 cmopn 12191 . . . . 5 class MetOpen
1311, 12cfv 5130 . . . 4 class (MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))))
145, 13wceq 1332 . . 3 wff (TopOpen‘𝑓) = (MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))))
15 ctps 12234 . . 3 class TopSp
1614, 2, 15crab 2421 . 2 class {𝑓 ∈ TopSp ∣ (TopOpen‘𝑓) = (MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))))}
171, 16wceq 1332 1 wff ∞MetSp = {𝑓 ∈ TopSp ∣ (TopOpen‘𝑓) = (MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))))}
Colors of variables: wff set class
This definition is referenced by:  isxms  12657
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