Definition df-fi 9381

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 9384). (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 9380	. 2 class fi
2		vx	. . 3 setvar 𝑥
3		cvv 3450	. . 3 class V
4		vz	. . . . . . 7 setvar 𝑧
5	4	cv 1569	. . . . . 6 class 𝑧
6		vy	. . . . . . . 8 setvar 𝑦
7	6	cv 1569	. . . . . . 7 class 𝑦
8	7	cint 4907	. . . . . 6 class ∩ 𝑦
9	5, 8	wceq 1570	. . . . 5 wff 𝑧 = ∩ 𝑦
10	2	cv 1569	. . . . . . 7 class 𝑥
11	10	cpw 4557	. . . . . 6 class 𝒫 𝑥
12		cfn 8952	. . . . . 6 class Fin
13	11, 12	cin 3898	. . . . 5 class (𝒫 𝑥 ∩ Fin)
14	9, 6, 13	wrex 3086	. . . 4 wff ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦
15	14, 4	cab 2738	. . 3 class {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦}
16	2, 3, 15	cmpt 5186	. 2 class (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})
17	1, 16	wceq 1570	1 wff fi = (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})

This definition is used by:  fival  9382