Definition df-fi 9301

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 9304). (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 9300	. 2 class fi
2		vx	. . 3 setvar 𝑥
3		cvv 3436	. . 3 class V
4		vz	. . . . . . 7 setvar 𝑧
5	4	cv 1539	. . . . . 6 class 𝑧
6		vy	. . . . . . . 8 setvar 𝑦
7	6	cv 1539	. . . . . . 7 class 𝑦
8	7	cint 4896	. . . . . 6 class ∩ 𝑦
9	5, 8	wceq 1540	. . . . 5 wff 𝑧 = ∩ 𝑦
10	2	cv 1539	. . . . . . 7 class 𝑥
11	10	cpw 4551	. . . . . 6 class 𝒫 𝑥
12		cfn 8872	. . . . . 6 class Fin
13	11, 12	cin 3902	. . . . 5 class (𝒫 𝑥 ∩ Fin)
14	9, 6, 13	wrex 3053	. . . 4 wff ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦
15	14, 4	cab 2707	. . 3 class {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦}
16	2, 3, 15	cmpt 5173	. 2 class (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})
17	1, 16	wceq 1540	1 wff fi = (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})

This definition is referenced by:  fival  9302