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Definition df-cms 24702
Description: Define the class of complete metric spaces. (Contributed by Mario Carneiro, 15-Oct-2015.)
Assertion
Ref Expression
df-cms CMetSp = {𝑀 ∈ MetSp ∣ [(Baseβ€˜π‘€) / 𝑏]((distβ€˜π‘€) β†Ύ (𝑏 Γ— 𝑏)) ∈ (CMetβ€˜π‘)}
Distinct variable group:   𝑀,𝑏

Detailed syntax breakdown of Definition df-cms
StepHypRef Expression
1 ccms 24699 . 2 class CMetSp
2 vw . . . . . . . 8 setvar 𝑀
32cv 1541 . . . . . . 7 class 𝑀
4 cds 17143 . . . . . . 7 class dist
53, 4cfv 6497 . . . . . 6 class (distβ€˜π‘€)
6 vb . . . . . . . 8 setvar 𝑏
76cv 1541 . . . . . . 7 class 𝑏
87, 7cxp 5632 . . . . . 6 class (𝑏 Γ— 𝑏)
95, 8cres 5636 . . . . 5 class ((distβ€˜π‘€) β†Ύ (𝑏 Γ— 𝑏))
10 ccmet 24621 . . . . . 6 class CMet
117, 10cfv 6497 . . . . 5 class (CMetβ€˜π‘)
129, 11wcel 2107 . . . 4 wff ((distβ€˜π‘€) β†Ύ (𝑏 Γ— 𝑏)) ∈ (CMetβ€˜π‘)
13 cbs 17084 . . . . 5 class Base
143, 13cfv 6497 . . . 4 class (Baseβ€˜π‘€)
1512, 6, 14wsbc 3740 . . 3 wff [(Baseβ€˜π‘€) / 𝑏]((distβ€˜π‘€) β†Ύ (𝑏 Γ— 𝑏)) ∈ (CMetβ€˜π‘)
16 cms 23674 . . 3 class MetSp
1715, 2, 16crab 3408 . 2 class {𝑀 ∈ MetSp ∣ [(Baseβ€˜π‘€) / 𝑏]((distβ€˜π‘€) β†Ύ (𝑏 Γ— 𝑏)) ∈ (CMetβ€˜π‘)}
181, 17wceq 1542 1 wff CMetSp = {𝑀 ∈ MetSp ∣ [(Baseβ€˜π‘€) / 𝑏]((distβ€˜π‘€) β†Ύ (𝑏 Γ— 𝑏)) ∈ (CMetβ€˜π‘)}
Colors of variables: wff setvar class
This definition is referenced by:  iscms  24712
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