Definition df-fi 8676

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 8679). (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 8675	. 2 class fi
2		vx	. . 3 setvar 𝑥
3		cvv 3417	. . 3 class V
4		vz	. . . . . . 7 setvar 𝑧
5	4	cv 1507	. . . . . 6 class 𝑧
6		vy	. . . . . . . 8 setvar 𝑦
7	6	cv 1507	. . . . . . 7 class 𝑦
8	7	cint 4754	. . . . . 6 class ∩ 𝑦
9	5, 8	wceq 1508	. . . . 5 wff 𝑧 = ∩ 𝑦
10	2	cv 1507	. . . . . . 7 class 𝑥
11	10	cpw 4425	. . . . . 6 class 𝒫 𝑥
12		cfn 8312	. . . . . 6 class Fin
13	11, 12	cin 3830	. . . . 5 class (𝒫 𝑥 ∩ Fin)
14	9, 6, 13	wrex 3091	. . . 4 wff ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦
15	14, 4	cab 2760	. . 3 class {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦}
16	2, 3, 15	cmpt 5013	. 2 class (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})
17	1, 16	wceq 1508	1 wff fi = (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})

This definition is referenced by:  fival  8677