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Definition df-tms 22174
Description: Define the function mapping a metric to a metric space. (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 ctmt 22171 . 2 class toMetSp
2 vd . . 3 setvar 𝑑
3 cxmt 19779 . . . . 5 class ∞Met
43crn 5144 . . . 4 class ran ∞Met
54cuni 4468 . . 3 class ran ∞Met
6 cnx 15901 . . . . . . 7 class ndx
7 cbs 15904 . . . . . . 7 class Base
86, 7cfv 5926 . . . . . 6 class (Base‘ndx)
92cv 1522 . . . . . . . 8 class 𝑑
109cdm 5143 . . . . . . 7 class dom 𝑑
1110cdm 5143 . . . . . 6 class dom dom 𝑑
128, 11cop 4216 . . . . 5 class ⟨(Base‘ndx), dom dom 𝑑
13 cds 15997 . . . . . . 7 class dist
146, 13cfv 5926 . . . . . 6 class (dist‘ndx)
1514, 9cop 4216 . . . . 5 class ⟨(dist‘ndx), 𝑑
1612, 15cpr 4212 . . . 4 class {⟨(Base‘ndx), dom dom 𝑑⟩, ⟨(dist‘ndx), 𝑑⟩}
17 cts 15994 . . . . . 6 class TopSet
186, 17cfv 5926 . . . . 5 class (TopSet‘ndx)
19 cmopn 19784 . . . . . 6 class MetOpen
209, 19cfv 5926 . . . . 5 class (MetOpen‘𝑑)
2118, 20cop 4216 . . . 4 class ⟨(TopSet‘ndx), (MetOpen‘𝑑)⟩
22 csts 15902 . . . 4 class sSet
2316, 21, 22co 6690 . . 3 class ({⟨(Base‘ndx), dom dom 𝑑⟩, ⟨(dist‘ndx), 𝑑⟩} sSet ⟨(TopSet‘ndx), (MetOpen‘𝑑)⟩)
242, 5, 23cmpt 4762 . 2 class (𝑑 ran ∞Met ↦ ({⟨(Base‘ndx), dom dom 𝑑⟩, ⟨(dist‘ndx), 𝑑⟩} sSet ⟨(TopSet‘ndx), (MetOpen‘𝑑)⟩))
251, 24wceq 1523 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  22333
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