Definition df-iin 3890

Description: Define indexed intersection. Definition of [Stoll] p. 45. See the remarks for its sibling operation of indexed union df-iun 3889. An alternate definition tying indexed intersection to ordinary intersection is dfiin2 3922. Theorem intiin 3942 provides a definition of ordinary intersection in terms of indexed intersection. (Contributed by NM, 27-Jun-1998.)

Assertion

Ref	Expression
df-iin	|- |^|_ x e. A B = { y | A. x e. A y e. B }

Distinct variable groups:   x, y   y, A   y, B

Allowed substitution hints:   A( x)   B( x)

Detailed syntax breakdown of Definition df-iin

Step	Hyp	Ref	Expression
1		vx	. . 3 setvar x
2		cA	. . 3 class A
3		cB	. . 3 class B
4	1, 2, 3	ciin 3888	. 2 class |^|_ x e. A B
5		vy	. . . . . 6 setvar y
6	5	cv 1352	. . . . 5 class y
7	6, 3	wcel 2148	. . . 4 wff y e. B
8	7, 1, 2	wral 2455	. . 3 wff A. x e. A y e. B
9	8, 5	cab 2163	. 2 class { y | A. x e. A y e. B }
10	4, 9	wceq 1353	1 wff |^|_ x e. A B = { y | A. x e. A y e. B }

This definition is referenced by:  eliin  3892  iineq1  3901  iineq2  3904  nfiinxy  3914  nfiinya  3916  nfii1  3918  dfiin2g  3920  cbviin  3925  intiin  3942  0iin  3946  viin  3947  iinxsng  3961  iinxprg  3962  iinuniss  3970  bdciin  14634