Definition df-topgen 12572

Description: Define a function that converts a basis to its corresponding topology. Equivalent to the definition of a topology generated by a basis in [Munkres] p. 78. (Contributed by NM, 16-Jul-2006.)

Assertion

Ref	Expression
df-topgen	⊢ topGen = (𝑥 ∈ V ↦ {𝑦 ∣ 𝑦 ⊆ ∪ (𝑥 ∩ 𝒫 𝑦)})

Distinct variable group:   𝑥,𝑦

Detailed syntax breakdown of Definition df-topgen

Step	Hyp	Ref	Expression
1		ctg 12566	. 2 class topGen
2		vx	. . 3 setvar 𝑥
3		cvv 2725	. . 3 class V
4		vy	. . . . . 6 setvar 𝑦
5	4	cv 1342	. . . . 5 class 𝑦
6	2	cv 1342	. . . . . . 7 class 𝑥
7	5	cpw 3558	. . . . . . 7 class 𝒫 𝑦
8	6, 7	cin 3114	. . . . . 6 class (𝑥 ∩ 𝒫 𝑦)
9	8	cuni 3788	. . . . 5 class ∪ (𝑥 ∩ 𝒫 𝑦)
10	5, 9	wss 3115	. . . 4 wff 𝑦 ⊆ ∪ (𝑥 ∩ 𝒫 𝑦)
11	10, 4	cab 2151	. . 3 class {𝑦 ∣ 𝑦 ⊆ ∪ (𝑥 ∩ 𝒫 𝑦)}
12	2, 3, 11	cmpt 4042	. 2 class (𝑥 ∈ V ↦ {𝑦 ∣ 𝑦 ⊆ ∪ (𝑥 ∩ 𝒫 𝑦)})
13	1, 12	wceq 1343	1 wff topGen = (𝑥 ∈ V ↦ {𝑦 ∣ 𝑦 ⊆ ∪ (𝑥 ∩ 𝒫 𝑦)})

This definition is referenced by:  tgval  12649  eltg4i  12655  eltg3  12657  tg1  12659  tg2  12660  tgclb  12665