Definition df-pnrm 23670

Description: Define perfectly normal spaces. A space is perfectly normal if it is normal and every closed set is a G_δ 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 ∈ (𝑗 ↑m ℕ) ↦ ∩ ran 𝑓)}

Distinct variable group:   𝑓,𝑗

Detailed syntax breakdown of Definition df-pnrm

Step	Hyp	Ref	Expression
1		cpnrm 23663	. 2 class PNrm
2		vj	. . . . . 6 setvar 𝑗
3	2	cv 1569	. . . . 5 class 𝑗
4		ccld 23367	. . . . 5 class Clsd
5	3, 4	cfv 6561	. . . 4 class (Clsd‘𝑗)
6		vf	. . . . . 6 setvar 𝑓
7		cn 12365	. . . . . . 7 class ℕ
8		cmap 8878	. . . . . . 7 class ↑m
9	3, 7, 8	co 7446	. . . . . 6 class (𝑗 ↑m ℕ)
10	6	cv 1569	. . . . . . . 8 class 𝑓
11	10	crn 5657	. . . . . . 7 class ran 𝑓
12	11	cint 4910	. . . . . 6 class ∩ ran 𝑓
13	6, 9, 12	cmpt 5190	. . . . 5 class (𝑓 ∈ (𝑗 ↑m ℕ) ↦ ∩ ran 𝑓)
14	13	crn 5657	. . . 4 class ran (𝑓 ∈ (𝑗 ↑m ℕ) ↦ ∩ ran 𝑓)
15	5, 14	wss 3900	. . 3 wff (Clsd‘𝑗) ⊆ ran (𝑓 ∈ (𝑗 ↑m ℕ) ↦ ∩ ran 𝑓)
16		cnrm 23661	. . 3 class Nrm
17	15, 2, 16	crab 3416	. 2 class {𝑗 ∈ Nrm ∣ (Clsd‘𝑗) ⊆ ran (𝑓 ∈ (𝑗 ↑m ℕ) ↦ ∩ ran 𝑓)}
18	1, 17	wceq 1570	1 wff PNrm = {𝑗 ∈ Nrm ∣ (Clsd‘𝑗) ⊆ ran (𝑓 ∈ (𝑗 ↑m ℕ) ↦ ∩ ran 𝑓)}

This definition is used by:  ispnrm  23690