Definition df-cmet 22863

Description: Define the class of complete metrics. (Contributed by Mario Carneiro, 1-May-2014.)

Assertion

Ref	Expression
df-cmet	⊢ CMet = (𝑥 ∈ V ↦ {𝑑 ∈ (Met‘𝑥) ∣ ∀𝑓 ∈ (CauFil‘𝑑)((MetOpen‘𝑑) fLim 𝑓) ≠ ∅})

Distinct variable group:   𝑓,𝑑,𝑥

Detailed syntax breakdown of Definition df-cmet

Step	Hyp	Ref	Expression
1		cms 22860	. 2 class CMet
2		vx	. . 3 setvar 𝑥
3		cvv 3173	. . 3 class V
4		vd	. . . . . . . . 9 setvar 𝑑
5	4	cv 1474	. . . . . . . 8 class 𝑑
6		cmopn 19557	. . . . . . . 8 class MetOpen
7	5, 6	cfv 5804	. . . . . . 7 class (MetOpen‘𝑑)
8		vf	. . . . . . . 8 setvar 𝑓
9	8	cv 1474	. . . . . . 7 class 𝑓
10		cflim 21548	. . . . . . 7 class fLim
11	7, 9, 10	co 6549	. . . . . 6 class ((MetOpen‘𝑑) fLim 𝑓)
12		c0 3874	. . . . . 6 class ∅
13	11, 12	wne 2780	. . . . 5 wff ((MetOpen‘𝑑) fLim 𝑓) ≠ ∅
14		ccfil 22858	. . . . . 6 class CauFil
15	5, 14	cfv 5804	. . . . 5 class (CauFil‘𝑑)
16	13, 8, 15	wral 2896	. . . 4 wff ∀𝑓 ∈ (CauFil‘𝑑)((MetOpen‘𝑑) fLim 𝑓) ≠ ∅
17	2	cv 1474	. . . . 5 class 𝑥
18		cme 19553	. . . . 5 class Met
19	17, 18	cfv 5804	. . . 4 class (Met‘𝑥)
20	16, 4, 19	crab 2900	. . 3 class {𝑑 ∈ (Met‘𝑥) ∣ ∀𝑓 ∈ (CauFil‘𝑑)((MetOpen‘𝑑) fLim 𝑓) ≠ ∅}
21	2, 3, 20	cmpt 4643	. 2 class (𝑥 ∈ V ↦ {𝑑 ∈ (Met‘𝑥) ∣ ∀𝑓 ∈ (CauFil‘𝑑)((MetOpen‘𝑑) fLim 𝑓) ≠ ∅})
22	1, 21	wceq 1475	1 wff CMet = (𝑥 ∈ V ↦ {𝑑 ∈ (Met‘𝑥) ∣ ∀𝑓 ∈ (CauFil‘𝑑)((MetOpen‘𝑑) fLim 𝑓) ≠ ∅})

This definition is referenced by:  iscmet  22890