Definition df-fi 7070

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 7073). (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 7069	. 2 class fi
2		vx	. . 3 setvar 𝑥
3		cvv 2771	. . 3 class V
4		vz	. . . . . . 7 setvar 𝑧
5	4	cv 1371	. . . . . 6 class 𝑧
6		vy	. . . . . . . 8 setvar 𝑦
7	6	cv 1371	. . . . . . 7 class 𝑦
8	7	cint 3884	. . . . . 6 class ∩ 𝑦
9	5, 8	wceq 1372	. . . . 5 wff 𝑧 = ∩ 𝑦
10	2	cv 1371	. . . . . . 7 class 𝑥
11	10	cpw 3615	. . . . . 6 class 𝒫 𝑥
12		cfn 6826	. . . . . 6 class Fin
13	11, 12	cin 3164	. . . . 5 class (𝒫 𝑥 ∩ Fin)
14	9, 6, 13	wrex 2484	. . . 4 wff ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦
15	14, 4	cab 2190	. . 3 class {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦}
16	2, 3, 15	cmpt 4104	. 2 class (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})
17	1, 16	wceq 1372	1 wff fi = (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})

This definition is referenced by:  fival  7071