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Definition df-cls 21039
Description: Define a function on topologies whose value is the closure function on the subsets of the base set. See clsval 21055. (Contributed by NM, 3-Oct-2006.)
Assertion
Ref Expression
df-cls cls = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 𝑗 {𝑦 ∈ (Clsd‘𝑗) ∣ 𝑥𝑦}))
Distinct variable group:   𝑥,𝑗,𝑦

Detailed syntax breakdown of Definition df-cls
StepHypRef Expression
1 ccl 21036 . 2 class cls
2 vj . . 3 setvar 𝑗
3 ctop 20911 . . 3 class Top
4 vx . . . 4 setvar 𝑥
52cv 1636 . . . . . 6 class 𝑗
65cuni 4630 . . . . 5 class 𝑗
76cpw 4351 . . . 4 class 𝒫 𝑗
84cv 1636 . . . . . . 7 class 𝑥
9 vy . . . . . . . 8 setvar 𝑦
109cv 1636 . . . . . . 7 class 𝑦
118, 10wss 3769 . . . . . 6 wff 𝑥𝑦
12 ccld 21034 . . . . . . 7 class Clsd
135, 12cfv 6101 . . . . . 6 class (Clsd‘𝑗)
1411, 9, 13crab 3100 . . . . 5 class {𝑦 ∈ (Clsd‘𝑗) ∣ 𝑥𝑦}
1514cint 4669 . . . 4 class {𝑦 ∈ (Clsd‘𝑗) ∣ 𝑥𝑦}
164, 7, 15cmpt 4923 . . 3 class (𝑥 ∈ 𝒫 𝑗 {𝑦 ∈ (Clsd‘𝑗) ∣ 𝑥𝑦})
172, 3, 16cmpt 4923 . 2 class (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 𝑗 {𝑦 ∈ (Clsd‘𝑗) ∣ 𝑥𝑦}))
181, 17wceq 1637 1 wff cls = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 𝑗 {𝑦 ∈ (Clsd‘𝑗) ∣ 𝑥𝑦}))
Colors of variables: wff setvar class
This definition is referenced by:  clsfval  21043
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