Definition df-fi 9435

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 9438). (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 9434	. 2 class fi
2		vx	. . 3 setvar 𝑥
3		cvv 3471	. . 3 class V
4		vz	. . . . . . 7 setvar 𝑧
5	4	cv 1533	. . . . . 6 class 𝑧
6		vy	. . . . . . . 8 setvar 𝑦
7	6	cv 1533	. . . . . . 7 class 𝑦
8	7	cint 4949	. . . . . 6 class ∩ 𝑦
9	5, 8	wceq 1534	. . . . 5 wff 𝑧 = ∩ 𝑦
10	2	cv 1533	. . . . . . 7 class 𝑥
11	10	cpw 4603	. . . . . 6 class 𝒫 𝑥
12		cfn 8964	. . . . . 6 class Fin
13	11, 12	cin 3946	. . . . 5 class (𝒫 𝑥 ∩ Fin)
14	9, 6, 13	wrex 3067	. . . 4 wff ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦
15	14, 4	cab 2705	. . 3 class {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦}
16	2, 3, 15	cmpt 5231	. 2 class (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})
17	1, 16	wceq 1534	1 wff fi = (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})

This definition is referenced by:  fival  9436