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Definition df-tms 22932
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
StepHypRef Expression
1 ctms 22929 . 2 class toMetSp
2 vd . . 3 setvar 𝑑
3 cxmet 20530 . . . . 5 class ∞Met
43crn 5543 . . . 4 class ran ∞Met
54cuni 4824 . . 3 class ran ∞Met
6 cnx 16480 . . . . . . 7 class ndx
7 cbs 16483 . . . . . . 7 class Base
86, 7cfv 6343 . . . . . 6 class (Base‘ndx)
92cv 1537 . . . . . . . 8 class 𝑑
109cdm 5542 . . . . . . 7 class dom 𝑑
1110cdm 5542 . . . . . 6 class dom dom 𝑑
128, 11cop 4556 . . . . 5 class ⟨(Base‘ndx), dom dom 𝑑
13 cds 16574 . . . . . . 7 class dist
146, 13cfv 6343 . . . . . 6 class (dist‘ndx)
1514, 9cop 4556 . . . . 5 class ⟨(dist‘ndx), 𝑑
1612, 15cpr 4552 . . . 4 class {⟨(Base‘ndx), dom dom 𝑑⟩, ⟨(dist‘ndx), 𝑑⟩}
17 cts 16571 . . . . . 6 class TopSet
186, 17cfv 6343 . . . . 5 class (TopSet‘ndx)
19 cmopn 20535 . . . . . 6 class MetOpen
209, 19cfv 6343 . . . . 5 class (MetOpen‘𝑑)
2118, 20cop 4556 . . . 4 class ⟨(TopSet‘ndx), (MetOpen‘𝑑)⟩
22 csts 16481 . . . 4 class sSet
2316, 21, 22co 7149 . . 3 class ({⟨(Base‘ndx), dom dom 𝑑⟩, ⟨(dist‘ndx), 𝑑⟩} sSet ⟨(TopSet‘ndx), (MetOpen‘𝑑)⟩)
242, 5, 23cmpt 5132 . 2 class (𝑑 ran ∞Met ↦ ({⟨(Base‘ndx), dom dom 𝑑⟩, ⟨(dist‘ndx), 𝑑⟩} sSet ⟨(TopSet‘ndx), (MetOpen‘𝑑)⟩))
251, 24wceq 1538 1 wff toMetSp = (𝑑 ran ∞Met ↦ ({⟨(Base‘ndx), dom dom 𝑑⟩, ⟨(dist‘ndx), 𝑑⟩} sSet ⟨(TopSet‘ndx), (MetOpen‘𝑑)⟩))
Colors of variables: wff setvar class
This definition is referenced by:  tmsval  23091
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