Definition df-xms 24532

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 24529	. 2 class ∞MetSp
2		vf	. . . . . 6 setvar 𝑓
3	2	cv 1569	. . . . 5 class 𝑓
4		ctopn 17500	. . . . 5 class TopOpen
5	3, 4	cfv 6540	. . . 4 class (TopOpen‘𝑓)
6		cds 17345	. . . . . . 7 class dist
7	3, 6	cfv 6540	. . . . . 6 class (dist‘𝑓)
8		cbs 17295	. . . . . . . 8 class Base
9	3, 8	cfv 6540	. . . . . . 7 class (Base‘𝑓)
10	9, 9	cxp 5661	. . . . . 6 class ((Base‘𝑓) × (Base‘𝑓))
11	7, 10	cres 5665	. . . . 5 class ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓)))
12		cmopn 21566	. . . . 5 class MetOpen
13	11, 12	cfv 6540	. . . 4 class (MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))))
14	5, 13	wceq 1570	. . 3 wff (TopOpen‘𝑓) = (MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))))
15		ctps 23143	. . 3 class TopSp
16	14, 2, 15	crab 3418	. 2 class {𝑓 ∈ TopSp ∣ (TopOpen‘𝑓) = (MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))))}
17	1, 16	wceq 1570	1 wff ∞MetSp = {𝑓 ∈ TopSp ∣ (TopOpen‘𝑓) = (MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))))}

This definition is used by:  isxms  24659