Definition df-fi 7269

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 7272). (Contributed by FL, 27-Apr-2008.)

Assertion

Ref	Expression
df-fi	|- fi = ( x e. _V |-> { z | E. y e. ( ~P x i^i Fin ) z = |^| y } )

Distinct variable group:   x, y, z

Detailed syntax breakdown of Definition df-fi

Step	Hyp	Ref	Expression
1		cfi 7268	. 2 class fi
2		vx	. . 3 setvar x
3		cvv 2815	. . 3 class _V
4		vz	. . . . . . 7 setvar z
5	4	cv 1397	. . . . . 6 class z
6		vy	. . . . . . . 8 setvar y
7	6	cv 1397	. . . . . . 7 class y
8	7	cint 3954	. . . . . 6 class |^| y
9	5, 8	wceq 1398	. . . . 5 wff z = |^| y
10	2	cv 1397	. . . . . . 7 class x
11	10	cpw 3674	. . . . . 6 class ~P x
12		cfn 6988	. . . . . 6 class Fin
13	11, 12	cin 3213	. . . . 5 class ( ~P x i^i Fin )
14	9, 6, 13	wrex 2523	. . . 4 wff E. y e. ( ~P x i^i Fin ) z = |^| y
15	14, 4	cab 2220	. . 3 class { z | E. y e. ( ~P x i^i Fin ) z = |^| y }
16	2, 3, 15	cmpt 4176	. 2 class ( x e. _V |-> { z | E. y e. ( ~P x i^i Fin ) z = |^| y } )
17	1, 16	wceq 1398	1 wff fi = ( x e. _V |-> { z | E. y e. ( ~P x i^i Fin ) z = |^| y } )

This definition is referenced by:  fival  7270