Definition df-iin 4523

Description: Define indexed intersection. Definition of [Stoll] p. 45. See the remarks for its sibling operation of indexed union df-iun 4522. An alternate definition tying indexed intersection to ordinary intersection is dfiin2 4555. Theorem intiin 4574 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 4521	. 2 class |^|_ x e. A B
5		vy	. . . . . 6 setvar y
6	5	cv 1482	. . . . 5 class y
7	6, 3	wcel 1990	. . . 4 wff y e. B
8	7, 1, 2	wral 2912	. . 3 wff A. x e. A y e. B
9	8, 5	cab 2608	. 2 class { y | A. x e. A y e. B }
10	4, 9	wceq 1483	1 wff |^|_ x e. A B = { y | A. x e. A y e. B }

This definition is referenced by:  eliin  4525  iineq1  4535  iineq2  4538  nfiin  4549  nfii1  4551  dfiin2g  4553  cbviin  4558  intiin  4574  0iin  4578  viin  4579  iinxsng  4600  iinxprg  4601  iinuni  4609  iineq12f  33973