Definition df-topsep 43915

Description: A topology is separable iff it has a countable dense subset. (Contributed by Stefan O'Rear, 8-Jan-2015.)

Assertion

Ref	Expression
df-topsep	⊢ TopSep = {𝑗 ∈ Top ∣ ∃𝑥 ∈ 𝒫 ∪ 𝑗(𝑥 ≼ ω ∧ ((cls‘𝑗)‘𝑥) = ∪ 𝑗)}

Distinct variable group:   𝑥,𝑗

Detailed syntax breakdown of Definition df-topsep

Step	Hyp	Ref	Expression
1		ctopsep 43913	. 2 class TopSep
2		vx	. . . . . . 7 setvar 𝑥
3	2	cv 1569	. . . . . 6 class 𝑥
4		com 7863	. . . . . 6 class ω
5		cdom 8942	. . . . . 6 class ≼
6	3, 4, 5	wbr 5110	. . . . 5 wff 𝑥 ≼ ω
7		vj	. . . . . . . . 9 setvar 𝑗
8	7	cv 1569	. . . . . . . 8 class 𝑗
9		ccl 23156	. . . . . . . 8 class cls
10	8, 9	cfv 6538	. . . . . . 7 class (cls‘𝑗)
11	3, 10	cfv 6538	. . . . . 6 class ((cls‘𝑗)‘𝑥)
12	8	cuni 4873	. . . . . 6 class ∪ 𝑗
13	11, 12	wceq 1570	. . . . 5 wff ((cls‘𝑗)‘𝑥) = ∪ 𝑗
14	6, 13	wa 400	. . . 4 wff (𝑥 ≼ ω ∧ ((cls‘𝑗)‘𝑥) = ∪ 𝑗)
15	12	cpw 4563	. . . 4 class 𝒫 ∪ 𝑗
16	14, 2, 15	wrex 3089	. . 3 wff ∃𝑥 ∈ 𝒫 ∪ 𝑗(𝑥 ≼ ω ∧ ((cls‘𝑗)‘𝑥) = ∪ 𝑗)
17		ctop 23031	. . 3 class Top
18	16, 7, 17	crab 3416	. 2 class {𝑗 ∈ Top ∣ ∃𝑥 ∈ 𝒫 ∪ 𝑗(𝑥 ≼ ω ∧ ((cls‘𝑗)‘𝑥) = ∪ 𝑗)}
19	1, 18	wceq 1570	1 wff TopSep = {𝑗 ∈ Top ∣ ∃𝑥 ∈ 𝒫 ∪ 𝑗(𝑥 ≼ ω ∧ ((cls‘𝑗)‘𝑥) = ∪ 𝑗)}

This definition is referenced by: (None)