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Mirrors > Home > MPE Home > Th. List > df-fi | Structured version Visualization version GIF version |
Description: Function whose value is the class of finite intersections of the elements of the argument. Note that the empty intersection being the universal class, hence a proper class, it cannot be an element of that class. Therefore, the function value is the class of nonempty finite intersections of elements of the argument (see elfi2 8867). (Contributed by FL, 27-Apr-2008.) |
Ref | Expression |
---|---|
df-fi | ⊢ fi = (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cfi 8863 | . 2 class fi | |
2 | vx | . . 3 setvar 𝑥 | |
3 | cvv 3495 | . . 3 class V | |
4 | vz | . . . . . . 7 setvar 𝑧 | |
5 | 4 | cv 1527 | . . . . . 6 class 𝑧 |
6 | vy | . . . . . . . 8 setvar 𝑦 | |
7 | 6 | cv 1527 | . . . . . . 7 class 𝑦 |
8 | 7 | cint 4869 | . . . . . 6 class ∩ 𝑦 |
9 | 5, 8 | wceq 1528 | . . . . 5 wff 𝑧 = ∩ 𝑦 |
10 | 2 | cv 1527 | . . . . . . 7 class 𝑥 |
11 | 10 | cpw 4537 | . . . . . 6 class 𝒫 𝑥 |
12 | cfn 8498 | . . . . . 6 class Fin | |
13 | 11, 12 | cin 3934 | . . . . 5 class (𝒫 𝑥 ∩ Fin) |
14 | 9, 6, 13 | wrex 3139 | . . . 4 wff ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦 |
15 | 14, 4 | cab 2799 | . . 3 class {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦} |
16 | 2, 3, 15 | cmpt 5138 | . 2 class (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦}) |
17 | 1, 16 | wceq 1528 | 1 wff fi = (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦}) |
Colors of variables: wff setvar class |
This definition is referenced by: fival 8865 |
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