Definition df-tms 22635

Description: Define the function mapping a metric to the metric space which it defines. (Contributed by Mario Carneiro, 2-Sep-2015.)

Assertion

Ref	Expression
df-tms	⊢ toMetSp = (𝑑 ∈ ∪ ran ∞Met ↦ ({⟨(Base‘ndx), dom dom 𝑑⟩, ⟨(dist‘ndx), 𝑑⟩} sSet ⟨(TopSet‘ndx), (MetOpen‘𝑑)⟩))

Detailed syntax breakdown of Definition df-tms

Step	Hyp	Ref	Expression
1		ctms 22632	. 2 class toMetSp
2		vd	. . 3 setvar 𝑑
3		cxmet 20232	. . . . 5 class ∞Met
4	3	crn 5408	. . . 4 class ran ∞Met
5	4	cuni 4712	. . 3 class ∪ ran ∞Met
6		cnx 16336	. . . . . . 7 class ndx
7		cbs 16339	. . . . . . 7 class Base
8	6, 7	cfv 6188	. . . . . 6 class (Base‘ndx)
9	2	cv 1506	. . . . . . . 8 class 𝑑
10	9	cdm 5407	. . . . . . 7 class dom 𝑑
11	10	cdm 5407	. . . . . 6 class dom dom 𝑑
12	8, 11	cop 4447	. . . . 5 class ⟨(Base‘ndx), dom dom 𝑑⟩
13		cds 16430	. . . . . . 7 class dist
14	6, 13	cfv 6188	. . . . . 6 class (dist‘ndx)
15	14, 9	cop 4447	. . . . 5 class ⟨(dist‘ndx), 𝑑⟩
16	12, 15	cpr 4443	. . . 4 class {⟨(Base‘ndx), dom dom 𝑑⟩, ⟨(dist‘ndx), 𝑑⟩}
17		cts 16427	. . . . . 6 class TopSet
18	6, 17	cfv 6188	. . . . 5 class (TopSet‘ndx)
19		cmopn 20237	. . . . . 6 class MetOpen
20	9, 19	cfv 6188	. . . . 5 class (MetOpen‘𝑑)
21	18, 20	cop 4447	. . . 4 class ⟨(TopSet‘ndx), (MetOpen‘𝑑)⟩
22		csts 16337	. . . 4 class sSet
23	16, 21, 22	co 6976	. . 3 class ({⟨(Base‘ndx), dom dom 𝑑⟩, ⟨(dist‘ndx), 𝑑⟩} sSet ⟨(TopSet‘ndx), (MetOpen‘𝑑)⟩)
24	2, 5, 23	cmpt 5008	. 2 class (𝑑 ∈ ∪ ran ∞Met ↦ ({⟨(Base‘ndx), dom dom 𝑑⟩, ⟨(dist‘ndx), 𝑑⟩} sSet ⟨(TopSet‘ndx), (MetOpen‘𝑑)⟩))
25	1, 24	wceq 1507	1 wff toMetSp = (𝑑 ∈ ∪ ran ∞Met ↦ ({⟨(Base‘ndx), dom dom 𝑑⟩, ⟨(dist‘ndx), 𝑑⟩} sSet ⟨(TopSet‘ndx), (MetOpen‘𝑑)⟩))

This definition is referenced by:  tmsval  22794