Definition df-fi 7097

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 7100). (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 7096	. 2 class fi
2		vx	. . 3 setvar 𝑥
3		cvv 2776	. . 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 3899	. . . . . 6 class ∩ 𝑦
9	5, 8	wceq 1373	. . . . 5 wff 𝑧 = ∩ 𝑦
10	2	cv 1372	. . . . . . 7 class 𝑥
11	10	cpw 3626	. . . . . 6 class 𝒫 𝑥
12		cfn 6850	. . . . . 6 class Fin
13	11, 12	cin 3173	. . . . 5 class (𝒫 𝑥 ∩ Fin)
14	9, 6, 13	wrex 2487	. . . 4 wff ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦
15	14, 4	cab 2193	. . 3 class {𝑧 ∣ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑧 = ∩ 𝑦}
16	2, 3, 15	cmpt 4121	. 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  7098