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Definition df-ms 22931
 Description: Define the (proper) class of metric spaces. (Contributed by NM, 27-Aug-2006.)
Assertion
Ref Expression
df-ms MetSp = {𝑓 ∈ ∞MetSp ∣ ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) ∈ (Met‘(Base‘𝑓))}

Detailed syntax breakdown of Definition df-ms
StepHypRef Expression
1 cms 22928 . 2 class MetSp
2 vf . . . . . . 7 setvar 𝑓
32cv 1537 . . . . . 6 class 𝑓
4 cds 16569 . . . . . 6 class dist
53, 4cfv 6328 . . . . 5 class (dist‘𝑓)
6 cbs 16478 . . . . . . 7 class Base
73, 6cfv 6328 . . . . . 6 class (Base‘𝑓)
87, 7cxp 5521 . . . . 5 class ((Base‘𝑓) × (Base‘𝑓))
95, 8cres 5525 . . . 4 class ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓)))
10 cmet 20080 . . . . 5 class Met
117, 10cfv 6328 . . . 4 class (Met‘(Base‘𝑓))
129, 11wcel 2112 . . 3 wff ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) ∈ (Met‘(Base‘𝑓))
13 cxms 22927 . . 3 class ∞MetSp
1412, 2, 13crab 3113 . 2 class {𝑓 ∈ ∞MetSp ∣ ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) ∈ (Met‘(Base‘𝑓))}
151, 14wceq 1538 1 wff MetSp = {𝑓 ∈ ∞MetSp ∣ ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) ∈ (Met‘(Base‘𝑓))}
 Colors of variables: wff setvar class This definition is referenced by:  isms  23059
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