Definition df-fi 8875

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 8878). (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 8874	. 2 class fi
2		vx	. . 3 setvar 𝑥
3		cvv 3494	. . 3 class V
4		vz	. . . . . . 7 setvar 𝑧
5	4	cv 1536	. . . . . 6 class 𝑧
6		vy	. . . . . . . 8 setvar 𝑦
7	6	cv 1536	. . . . . . 7 class 𝑦
8	7	cint 4876	. . . . . 6 class ∩ 𝑦
9	5, 8	wceq 1537	. . . . 5 wff 𝑧 = ∩ 𝑦
10	2	cv 1536	. . . . . . 7 class 𝑥
11	10	cpw 4539	. . . . . 6 class 𝒫 𝑥
12		cfn 8509	. . . . . 6 class Fin
13	11, 12	cin 3935	. . . . 5 class (𝒫 𝑥 ∩ Fin)
14	9, 6, 13	wrex 3139	. . . 4 wff ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦
15	14, 4	cab 2799	. . 3 class {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦}
16	2, 3, 15	cmpt 5146	. 2 class (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})
17	1, 16	wceq 1537	1 wff fi = (𝑥 ∈ V ↦ {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦})

This definition is referenced by:  fival  8876