Definition df-fi 7159

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 7162). (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 7158	. 2 class fi
2		vx	. . 3 setvar 𝑥
3		cvv 2800	. . 3 class V
4		vz	. . . . . . 7 setvar 𝑧
5	4	cv 1394	. . . . . 6 class 𝑧
6		vy	. . . . . . . 8 setvar 𝑦
7	6	cv 1394	. . . . . . 7 class 𝑦
8	7	cint 3926	. . . . . 6 class ∩ 𝑦
9	5, 8	wceq 1395	. . . . 5 wff 𝑧 = ∩ 𝑦
10	2	cv 1394	. . . . . . 7 class 𝑥
11	10	cpw 3650	. . . . . 6 class 𝒫 𝑥
12		cfn 6904	. . . . . 6 class Fin
13	11, 12	cin 3197	. . . . 5 class (𝒫 𝑥 ∩ Fin)
14	9, 6, 13	wrex 2509	. . . 4 wff ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦
15	14, 4	cab 2215	. . 3 class {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦}
16	2, 3, 15	cmpt 4148	. 2 class (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})
17	1, 16	wceq 1395	1 wff fi = (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})

This definition is referenced by:  fival  7160