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Definition df-metu 19947
Description: Define the function mapping metrics to the uniform structure generated by that metric. (Contributed by Thierry Arnoux, 1-Dec-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
Assertion
Ref Expression
df-metu metUnif = (𝑑 ran PsMet ↦ ((dom dom 𝑑 × dom dom 𝑑)filGenran (𝑎 ∈ ℝ+ ↦ (𝑑 “ (0[,)𝑎)))))
Distinct variable group:   𝑎,𝑑

Detailed syntax breakdown of Definition df-metu
StepHypRef Expression
1 cmetu 19939 . 2 class metUnif
2 vd . . 3 setvar 𝑑
3 cpsmet 19932 . . . . 5 class PsMet
43crn 5267 . . . 4 class ran PsMet
54cuni 4588 . . 3 class ran PsMet
62cv 1631 . . . . . . 7 class 𝑑
76cdm 5266 . . . . . 6 class dom 𝑑
87cdm 5266 . . . . 5 class dom dom 𝑑
98, 8cxp 5264 . . . 4 class (dom dom 𝑑 × dom dom 𝑑)
10 va . . . . . 6 setvar 𝑎
11 crp 12025 . . . . . 6 class +
126ccnv 5265 . . . . . . 7 class 𝑑
13 cc0 10128 . . . . . . . 8 class 0
1410cv 1631 . . . . . . . 8 class 𝑎
15 cico 12370 . . . . . . . 8 class [,)
1613, 14, 15co 6813 . . . . . . 7 class (0[,)𝑎)
1712, 16cima 5269 . . . . . 6 class (𝑑 “ (0[,)𝑎))
1810, 11, 17cmpt 4881 . . . . 5 class (𝑎 ∈ ℝ+ ↦ (𝑑 “ (0[,)𝑎)))
1918crn 5267 . . . 4 class ran (𝑎 ∈ ℝ+ ↦ (𝑑 “ (0[,)𝑎)))
20 cfg 19937 . . . 4 class filGen
219, 19, 20co 6813 . . 3 class ((dom dom 𝑑 × dom dom 𝑑)filGenran (𝑎 ∈ ℝ+ ↦ (𝑑 “ (0[,)𝑎))))
222, 5, 21cmpt 4881 . 2 class (𝑑 ran PsMet ↦ ((dom dom 𝑑 × dom dom 𝑑)filGenran (𝑎 ∈ ℝ+ ↦ (𝑑 “ (0[,)𝑎)))))
231, 22wceq 1632 1 wff metUnif = (𝑑 ran PsMet ↦ ((dom dom 𝑑 × dom dom 𝑑)filGenran (𝑎 ∈ ℝ+ ↦ (𝑑 “ (0[,)𝑎)))))
Colors of variables: wff setvar class
This definition is referenced by:  metuval  22555
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