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Definition df-reg 21030
Description: Define regular spaces. A space is regular if a point and a closed set can be separated by neighborhoods. (Contributed by Jeff Hankins, 1-Feb-2010.)
Assertion
Ref Expression
df-reg Reg = {𝑗 ∈ Top ∣ ∀𝑥𝑗𝑦𝑥𝑧𝑗 (𝑦𝑧 ∧ ((cls‘𝑗)‘𝑧) ⊆ 𝑥)}
Distinct variable group:   𝑥,𝑗,𝑦,𝑧

Detailed syntax breakdown of Definition df-reg
StepHypRef Expression
1 creg 21023 . 2 class Reg
2 vy . . . . . . . 8 setvar 𝑦
3 vz . . . . . . . 8 setvar 𝑧
42, 3wel 1988 . . . . . . 7 wff 𝑦𝑧
53cv 1479 . . . . . . . . 9 class 𝑧
6 vj . . . . . . . . . . 11 setvar 𝑗
76cv 1479 . . . . . . . . . 10 class 𝑗
8 ccl 20732 . . . . . . . . . 10 class cls
97, 8cfv 5847 . . . . . . . . 9 class (cls‘𝑗)
105, 9cfv 5847 . . . . . . . 8 class ((cls‘𝑗)‘𝑧)
11 vx . . . . . . . . 9 setvar 𝑥
1211cv 1479 . . . . . . . 8 class 𝑥
1310, 12wss 3555 . . . . . . 7 wff ((cls‘𝑗)‘𝑧) ⊆ 𝑥
144, 13wa 384 . . . . . 6 wff (𝑦𝑧 ∧ ((cls‘𝑗)‘𝑧) ⊆ 𝑥)
1514, 3, 7wrex 2908 . . . . 5 wff 𝑧𝑗 (𝑦𝑧 ∧ ((cls‘𝑗)‘𝑧) ⊆ 𝑥)
1615, 2, 12wral 2907 . . . 4 wff 𝑦𝑥𝑧𝑗 (𝑦𝑧 ∧ ((cls‘𝑗)‘𝑧) ⊆ 𝑥)
1716, 11, 7wral 2907 . . 3 wff 𝑥𝑗𝑦𝑥𝑧𝑗 (𝑦𝑧 ∧ ((cls‘𝑗)‘𝑧) ⊆ 𝑥)
18 ctop 20617 . . 3 class Top
1917, 6, 18crab 2911 . 2 class {𝑗 ∈ Top ∣ ∀𝑥𝑗𝑦𝑥𝑧𝑗 (𝑦𝑧 ∧ ((cls‘𝑗)‘𝑧) ⊆ 𝑥)}
201, 19wceq 1480 1 wff Reg = {𝑗 ∈ Top ∣ ∀𝑥𝑗𝑦𝑥𝑧𝑗 (𝑦𝑧 ∧ ((cls‘𝑗)‘𝑧) ⊆ 𝑥)}
Colors of variables: wff setvar class
This definition is referenced by:  isreg  21046
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