Definition df-xms 12508

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

Step	Hyp	Ref	Expression
1		cxms 12505	. 2 class ∞MetSp
2		vf	. . . . . 6 setvar 𝑓
3	2	cv 1330	. . . . 5 class 𝑓
4		ctopn 12121	. . . . 5 class TopOpen
5	3, 4	cfv 5123	. . . 4 class (TopOpen‘𝑓)
6		cds 12030	. . . . . . 7 class dist
7	3, 6	cfv 5123	. . . . . 6 class (dist‘𝑓)
8		cbs 11959	. . . . . . . 8 class Base
9	3, 8	cfv 5123	. . . . . . 7 class (Base‘𝑓)
10	9, 9	cxp 4537	. . . . . 6 class ((Base‘𝑓) × (Base‘𝑓))
11	7, 10	cres 4541	. . . . 5 class ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓)))
12		cmopn 12154	. . . . 5 class MetOpen
13	11, 12	cfv 5123	. . . 4 class (MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))))
14	5, 13	wceq 1331	. . 3 wff (TopOpen‘𝑓) = (MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))))
15		ctps 12197	. . 3 class TopSp
16	14, 2, 15	crab 2420	. 2 class {𝑓 ∈ TopSp ∣ (TopOpen‘𝑓) = (MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))))}
17	1, 16	wceq 1331	1 wff ∞MetSp = {𝑓 ∈ TopSp ∣ (TopOpen‘𝑓) = (MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))))}

This definition is referenced by:  isxms  12620