Definition df-fi 9316

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 9319). (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 9315	. 2 class fi
2		vx	. . 3 setvar 𝑥
3		cvv 3440	. . 3 class V
4		vz	. . . . . . 7 setvar 𝑧
5	4	cv 1540	. . . . . 6 class 𝑧
6		vy	. . . . . . . 8 setvar 𝑦
7	6	cv 1540	. . . . . . 7 class 𝑦
8	7	cint 4902	. . . . . 6 class ∩ 𝑦
9	5, 8	wceq 1541	. . . . 5 wff 𝑧 = ∩ 𝑦
10	2	cv 1540	. . . . . . 7 class 𝑥
11	10	cpw 4554	. . . . . 6 class 𝒫 𝑥
12		cfn 8885	. . . . . 6 class Fin
13	11, 12	cin 3900	. . . . 5 class (𝒫 𝑥 ∩ Fin)
14	9, 6, 13	wrex 3060	. . . 4 wff ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦
15	14, 4	cab 2714	. . 3 class {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦}
16	2, 3, 15	cmpt 5179	. 2 class (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})
17	1, 16	wceq 1541	1 wff fi = (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})

This definition is referenced by:  fival  9317