Definition df-fi 7028

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 7031). (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 7027	. 2 class fi
2		vx	. . 3 setvar 𝑥
3		cvv 2760	. . 3 class V
4		vz	. . . . . . 7 setvar 𝑧
5	4	cv 1363	. . . . . 6 class 𝑧
6		vy	. . . . . . . 8 setvar 𝑦
7	6	cv 1363	. . . . . . 7 class 𝑦
8	7	cint 3870	. . . . . 6 class ∩ 𝑦
9	5, 8	wceq 1364	. . . . 5 wff 𝑧 = ∩ 𝑦
10	2	cv 1363	. . . . . . 7 class 𝑥
11	10	cpw 3601	. . . . . 6 class 𝒫 𝑥
12		cfn 6794	. . . . . 6 class Fin
13	11, 12	cin 3152	. . . . 5 class (𝒫 𝑥 ∩ Fin)
14	9, 6, 13	wrex 2473	. . . 4 wff ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦
15	14, 4	cab 2179	. . 3 class {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦}
16	2, 3, 15	cmpt 4090	. 2 class (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})
17	1, 16	wceq 1364	1 wff fi = (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})

This definition is referenced by:  fival  7029