Definition df-fi 9369

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 9372). (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 9368	. 2 class fi
2		vx	. . 3 setvar 𝑥
3		cvv 3450	. . 3 class V
4		vz	. . . . . . 7 setvar 𝑧
5	4	cv 1539	. . . . . 6 class 𝑧
6		vy	. . . . . . . 8 setvar 𝑦
7	6	cv 1539	. . . . . . 7 class 𝑦
8	7	cint 4913	. . . . . 6 class ∩ 𝑦
9	5, 8	wceq 1540	. . . . 5 wff 𝑧 = ∩ 𝑦
10	2	cv 1539	. . . . . . 7 class 𝑥
11	10	cpw 4566	. . . . . 6 class 𝒫 𝑥
12		cfn 8921	. . . . . 6 class Fin
13	11, 12	cin 3916	. . . . 5 class (𝒫 𝑥 ∩ Fin)
14	9, 6, 13	wrex 3054	. . . 4 wff ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦
15	14, 4	cab 2708	. . 3 class {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦}
16	2, 3, 15	cmpt 5191	. 2 class (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})
17	1, 16	wceq 1540	1 wff fi = (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})

This definition is referenced by:  fival  9370