Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > df-cref | Structured version Visualization version GIF version |
Description: Define a statement "every open cover has an 𝐴 refinement" , where 𝐴 is a property for refinements like "finite", "countable", "point finite" or "locally finite". (Contributed by Thierry Arnoux, 7-Jan-2020.) |
Ref | Expression |
---|---|
df-cref | ⊢ CovHasRef𝐴 = {𝑗 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑗(∪ 𝑗 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑗 ∩ 𝐴)𝑧Ref𝑦)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cA | . . 3 class 𝐴 | |
2 | 1 | ccref 29237 | . 2 class CovHasRef𝐴 |
3 | vj | . . . . . . . 8 setvar 𝑗 | |
4 | 3 | cv 1474 | . . . . . . 7 class 𝑗 |
5 | 4 | cuni 4372 | . . . . . 6 class ∪ 𝑗 |
6 | vy | . . . . . . . 8 setvar 𝑦 | |
7 | 6 | cv 1474 | . . . . . . 7 class 𝑦 |
8 | 7 | cuni 4372 | . . . . . 6 class ∪ 𝑦 |
9 | 5, 8 | wceq 1475 | . . . . 5 wff ∪ 𝑗 = ∪ 𝑦 |
10 | vz | . . . . . . . 8 setvar 𝑧 | |
11 | 10 | cv 1474 | . . . . . . 7 class 𝑧 |
12 | cref 21115 | . . . . . . 7 class Ref | |
13 | 11, 7, 12 | wbr 4583 | . . . . . 6 wff 𝑧Ref𝑦 |
14 | 4 | cpw 4108 | . . . . . . 7 class 𝒫 𝑗 |
15 | 14, 1 | cin 3539 | . . . . . 6 class (𝒫 𝑗 ∩ 𝐴) |
16 | 13, 10, 15 | wrex 2897 | . . . . 5 wff ∃𝑧 ∈ (𝒫 𝑗 ∩ 𝐴)𝑧Ref𝑦 |
17 | 9, 16 | wi 4 | . . . 4 wff (∪ 𝑗 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑗 ∩ 𝐴)𝑧Ref𝑦) |
18 | 17, 6, 14 | wral 2896 | . . 3 wff ∀𝑦 ∈ 𝒫 𝑗(∪ 𝑗 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑗 ∩ 𝐴)𝑧Ref𝑦) |
19 | ctop 20517 | . . 3 class Top | |
20 | 18, 3, 19 | crab 2900 | . 2 class {𝑗 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑗(∪ 𝑗 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑗 ∩ 𝐴)𝑧Ref𝑦)} |
21 | 2, 20 | wceq 1475 | 1 wff CovHasRef𝐴 = {𝑗 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑗(∪ 𝑗 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑗 ∩ 𝐴)𝑧Ref𝑦)} |
Colors of variables: wff setvar class |
This definition is referenced by: iscref 29239 crefeq 29240 |
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