Definition df-fi 7071

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 7074). (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 7070	. 2 class fi
2		vx	. . 3 setvar 𝑥
3		cvv 2772	. . 3 class V
4		vz	. . . . . . 7 setvar 𝑧
5	4	cv 1372	. . . . . 6 class 𝑧
6		vy	. . . . . . . 8 setvar 𝑦
7	6	cv 1372	. . . . . . 7 class 𝑦
8	7	cint 3885	. . . . . 6 class ∩ 𝑦
9	5, 8	wceq 1373	. . . . 5 wff 𝑧 = ∩ 𝑦
10	2	cv 1372	. . . . . . 7 class 𝑥
11	10	cpw 3616	. . . . . 6 class 𝒫 𝑥
12		cfn 6827	. . . . . 6 class Fin
13	11, 12	cin 3165	. . . . 5 class (𝒫 𝑥 ∩ Fin)
14	9, 6, 13	wrex 2485	. . . 4 wff ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦
15	14, 4	cab 2191	. . 3 class {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦}
16	2, 3, 15	cmpt 4105	. 2 class (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})
17	1, 16	wceq 1373	1 wff fi = (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})

This definition is referenced by:  fival  7072