Definition df-fi 7044

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 7047). (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 7043	. 2 class fi
2		vx	. . 3 setvar 𝑥
3		cvv 2763	. . 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 3875	. . . . . 6 class ∩ 𝑦
9	5, 8	wceq 1364	. . . . 5 wff 𝑧 = ∩ 𝑦
10	2	cv 1363	. . . . . . 7 class 𝑥
11	10	cpw 3606	. . . . . 6 class 𝒫 𝑥
12		cfn 6808	. . . . . 6 class Fin
13	11, 12	cin 3156	. . . . 5 class (𝒫 𝑥 ∩ Fin)
14	9, 6, 13	wrex 2476	. . . 4 wff ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦
15	14, 4	cab 2182	. . 3 class {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦}
16	2, 3, 15	cmpt 4095	. 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  7045