Definition df-fi 9367

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 9370). (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 9366	. 2 class fi
2		vx	. . 3 setvar 𝑥
3		cvv 3455	. . 3 class V
4		vz	. . . . . . 7 setvar 𝑧
5	4	cv 1569	. . . . . 6 class 𝑧
6		vy	. . . . . . . 8 setvar 𝑦
7	6	cv 1569	. . . . . . 7 class 𝑦
8	7	cint 4912	. . . . . 6 class ∩ 𝑦
9	5, 8	wceq 1570	. . . . 5 wff 𝑧 = ∩ 𝑦
10	2	cv 1569	. . . . . . 7 class 𝑥
11	10	cpw 4562	. . . . . 6 class 𝒫 𝑥
12		cfn 8939	. . . . . 6 class Fin
13	11, 12	cin 3904	. . . . 5 class (𝒫 𝑥 ∩ Fin)
14	9, 6, 13	wrex 3089	. . . 4 wff ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦
15	14, 4	cab 2741	. . 3 class {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦}
16	2, 3, 15	cmpt 5192	. 2 class (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})
17	1, 16	wceq 1570	1 wff fi = (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})

This definition is used by:  fival  9368