Definition df-fi 9403

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 9406). (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 9402	. 2 class fi
2		vx	. . 3 setvar 𝑥
3		cvv 3466	. . 3 class V
4		vz	. . . . . . 7 setvar 𝑧
5	4	cv 1532	. . . . . 6 class 𝑧
6		vy	. . . . . . . 8 setvar 𝑦
7	6	cv 1532	. . . . . . 7 class 𝑦
8	7	cint 4941	. . . . . 6 class ∩ 𝑦
9	5, 8	wceq 1533	. . . . 5 wff 𝑧 = ∩ 𝑦
10	2	cv 1532	. . . . . . 7 class 𝑥
11	10	cpw 4595	. . . . . 6 class 𝒫 𝑥
12		cfn 8936	. . . . . 6 class Fin
13	11, 12	cin 3940	. . . . 5 class (𝒫 𝑥 ∩ Fin)
14	9, 6, 13	wrex 3062	. . . 4 wff ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦
15	14, 4	cab 2701	. . 3 class {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦}
16	2, 3, 15	cmpt 5222	. 2 class (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})
17	1, 16	wceq 1533	1 wff fi = (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})

This definition is referenced by:  fival  9404