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Definition df-xms 21935
 Description: Define the (proper) class of all 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 cxme 21932 . 2 class ∞MetSp
2 vf . . . . . 6 setvar 𝑓
32cv 1474 . . . . 5 class 𝑓
4 ctopn 15905 . . . . 5 class TopOpen
53, 4cfv 5804 . . . 4 class (TopOpen‘𝑓)
6 cds 15777 . . . . . . 7 class dist
73, 6cfv 5804 . . . . . 6 class (dist‘𝑓)
8 cbs 15695 . . . . . . . 8 class Base
93, 8cfv 5804 . . . . . . 7 class (Base‘𝑓)
109, 9cxp 5036 . . . . . 6 class ((Base‘𝑓) × (Base‘𝑓))
117, 10cres 5040 . . . . 5 class ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓)))
12 cmopn 19557 . . . . 5 class MetOpen
1311, 12cfv 5804 . . . 4 class (MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))))
145, 13wceq 1475 . . 3 wff (TopOpen‘𝑓) = (MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))))
15 ctps 20519 . . 3 class TopSp
1614, 2, 15crab 2900 . 2 class {𝑓 ∈ TopSp ∣ (TopOpen‘𝑓) = (MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))))}
171, 16wceq 1475 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  22062
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