Definition df-fi 6865

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 6868). (Contributed by FL, 27-Apr-2008.)

Assertion

Ref	Expression
df-fi	⊢ fi = (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})

Distinct variable group:   𝑥,𝑦,𝑧

Detailed syntax breakdown of Definition df-fi

Step	Hyp	Ref	Expression
1		cfi 6864	. 2 class fi
2		vx	. . 3 setvar 𝑥
3		cvv 2689	. . 3 class V
4		vz	. . . . . . 7 setvar 𝑧
5	4	cv 1331	. . . . . 6 class 𝑧
6		vy	. . . . . . . 8 setvar 𝑦
7	6	cv 1331	. . . . . . 7 class 𝑦
8	7	cint 3779	. . . . . 6 class ∩ 𝑦
9	5, 8	wceq 1332	. . . . 5 wff 𝑧 = ∩ 𝑦
10	2	cv 1331	. . . . . . 7 class 𝑥
11	10	cpw 3515	. . . . . 6 class 𝒫 𝑥
12		cfn 6642	. . . . . 6 class Fin
13	11, 12	cin 3075	. . . . 5 class (𝒫 𝑥 ∩ Fin)
14	9, 6, 13	wrex 2418	. . . 4 wff ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦
15	14, 4	cab 2126	. . 3 class {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦}
16	2, 3, 15	cmpt 3997	. 2 class (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})
17	1, 16	wceq 1332	1 wff fi = (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})

This definition is referenced by:  fival  6866