Definition df-iin 3852

Description: Define indexed intersection. Definition of [Stoll] p. 45. See the remarks for its sibling operation of indexed union df-iun 3851. An alternate definition tying indexed intersection to ordinary intersection is dfiin2 3884. Theorem intiin 3903 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 3850	. 2 class |^|_ x e. A B
5		vy	. . . . . 6 setvar y
6	5	cv 1334	. . . . 5 class y
7	6, 3	wcel 2128	. . . 4 wff y e. B
8	7, 1, 2	wral 2435	. . 3 wff A. x e. A y e. B
9	8, 5	cab 2143	. 2 class { y | A. x e. A y e. B }
10	4, 9	wceq 1335	1 wff |^|_ x e. A B = { y | A. x e. A y e. B }

This definition is referenced by:  eliin  3854  iineq1  3863  iineq2  3866  nfiinxy  3876  nfiinya  3878  nfii1  3880  dfiin2g  3882  cbviin  3887  intiin  3903  0iin  3907  viin  3908  iinxsng  3922  iinxprg  3923  iinuniss  3931  bdciin  13425