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Mirrors > Home > ILE Home > Th. List > df-topgen | GIF version |
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.) |
Ref | Expression |
---|---|
df-topgen | ⊢ topGen = (𝑥 ∈ V ↦ {𝑦 ∣ 𝑦 ⊆ ∪ (𝑥 ∩ 𝒫 𝑦)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ctg 12721 | . 2 class topGen | |
2 | vx | . . 3 setvar 𝑥 | |
3 | cvv 2749 | . . 3 class V | |
4 | vy | . . . . . 6 setvar 𝑦 | |
5 | 4 | cv 1362 | . . . . 5 class 𝑦 |
6 | 2 | cv 1362 | . . . . . . 7 class 𝑥 |
7 | 5 | cpw 3587 | . . . . . . 7 class 𝒫 𝑦 |
8 | 6, 7 | cin 3140 | . . . . . 6 class (𝑥 ∩ 𝒫 𝑦) |
9 | 8 | cuni 3821 | . . . . 5 class ∪ (𝑥 ∩ 𝒫 𝑦) |
10 | 5, 9 | wss 3141 | . . . 4 wff 𝑦 ⊆ ∪ (𝑥 ∩ 𝒫 𝑦) |
11 | 10, 4 | cab 2173 | . . 3 class {𝑦 ∣ 𝑦 ⊆ ∪ (𝑥 ∩ 𝒫 𝑦)} |
12 | 2, 3, 11 | cmpt 4076 | . 2 class (𝑥 ∈ V ↦ {𝑦 ∣ 𝑦 ⊆ ∪ (𝑥 ∩ 𝒫 𝑦)}) |
13 | 1, 12 | wceq 1363 | 1 wff topGen = (𝑥 ∈ V ↦ {𝑦 ∣ 𝑦 ⊆ ∪ (𝑥 ∩ 𝒫 𝑦)}) |
Colors of variables: wff set class |
This definition is referenced by: tgval 12729 eltg4i 13908 eltg3 13910 tg1 13912 tg2 13913 tgclb 13918 |
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