Definition df-iin 3919

Description: Define indexed intersection. Definition of [Stoll] p. 45. See the remarks for its sibling operation of indexed union df-iun 3918. An alternate definition tying indexed intersection to ordinary intersection is dfiin2 3951. Theorem intiin 3971 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 3917	. 2 class |^|_ x e. A B
5		vy	. . . . . 6 setvar y
6	5	cv 1363	. . . . 5 class y
7	6, 3	wcel 2167	. . . 4 wff y e. B
8	7, 1, 2	wral 2475	. . 3 wff A. x e. A y e. B
9	8, 5	cab 2182	. 2 class { y | A. x e. A y e. B }
10	4, 9	wceq 1364	1 wff |^|_ x e. A B = { y | A. x e. A y e. B }

This definition is referenced by:  eliin  3921  iineq1  3930  iineq2  3933  nfiinxy  3943  nfiinya  3945  nfii1  3947  dfiin2g  3949  cbviin  3954  intiin  3971  0iin  3975  viin  3976  iinxsng  3990  iinxprg  3991  iinuniss  3999  bdciin  15525