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Definition df-pnrm 21117
Description: Define perfectly normal spaces. A space is perfectly normal if it is normal and every closed set is a G&delta; set, meaning that it is a countable intersection of open sets. (Contributed by Mario Carneiro, 26-Aug-2015.)
Assertion
Ref Expression
df-pnrm PNrm = {𝑗 ∈ Nrm ∣ (Clsd‘𝑗) ⊆ ran (𝑓 ∈ (𝑗𝑚 ℕ) ↦ ran 𝑓)}
Distinct variable group:   𝑓,𝑗

Detailed syntax breakdown of Definition df-pnrm
StepHypRef Expression
1 cpnrm 21110 . 2 class PNrm
2 vj . . . . . 6 setvar 𝑗
32cv 1481 . . . . 5 class 𝑗
4 ccld 20814 . . . . 5 class Clsd
53, 4cfv 5886 . . . 4 class (Clsd‘𝑗)
6 vf . . . . . 6 setvar 𝑓
7 cn 11017 . . . . . . 7 class
8 cmap 7854 . . . . . . 7 class 𝑚
93, 7, 8co 6647 . . . . . 6 class (𝑗𝑚 ℕ)
106cv 1481 . . . . . . . 8 class 𝑓
1110crn 5113 . . . . . . 7 class ran 𝑓
1211cint 4473 . . . . . 6 class ran 𝑓
136, 9, 12cmpt 4727 . . . . 5 class (𝑓 ∈ (𝑗𝑚 ℕ) ↦ ran 𝑓)
1413crn 5113 . . . 4 class ran (𝑓 ∈ (𝑗𝑚 ℕ) ↦ ran 𝑓)
155, 14wss 3572 . . 3 wff (Clsd‘𝑗) ⊆ ran (𝑓 ∈ (𝑗𝑚 ℕ) ↦ ran 𝑓)
16 cnrm 21108 . . 3 class Nrm
1715, 2, 16crab 2915 . 2 class {𝑗 ∈ Nrm ∣ (Clsd‘𝑗) ⊆ ran (𝑓 ∈ (𝑗𝑚 ℕ) ↦ ran 𝑓)}
181, 17wceq 1482 1 wff PNrm = {𝑗 ∈ Nrm ∣ (Clsd‘𝑗) ⊆ ran (𝑓 ∈ (𝑗𝑚 ℕ) ↦ ran 𝑓)}
Colors of variables: wff setvar class
This definition is referenced by:  ispnrm  21137
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