Definition df-iin 3973

Description: Define indexed intersection. Definition of [Stoll] p. 45. See the remarks for its sibling operation of indexed union df-iun 3972. An alternate definition tying indexed intersection to ordinary intersection is dfiin2 4003. Theorem intiin 4021 provides a definition of ordinary intersection in terms of indexed intersection. (Contributed by NM, 27-Jun-1998.)

Assertion

Ref	Expression
df-iin	⊢ ∩x ∈ A B = {y ∣ ∀x ∈ A y ∈ 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 3971	. 2 class ∩x ∈ A B
5		vy	. . . . . 6 setvar y
6	5	cv 1641	. . . . 5 class y
7	6, 3	wcel 1710	. . . 4 wff y ∈ B
8	7, 1, 2	wral 2615	. . 3 wff ∀x ∈ A y ∈ B
9	8, 5	cab 2339	. 2 class {y ∣ ∀x ∈ A y ∈ B}
10	4, 9	wceq 1642	1 wff ∩x ∈ A B = {y ∣ ∀x ∈ A y ∈ B}

This definition is referenced by:  eliin  3975  iineq1  3984  iineq2  3987  nfiin  3997  nfii1  3999  dfiin2g  4001  cbviin  4005  intiin  4021  0iin  4025  viin  4026  iinxsng  4043