Definition df-iin 3947

Description: Define indexed intersection. Definition of [Stoll] p. 45. See the remarks for its sibling operation of indexed union df-iun 3946. An alternate definition tying indexed intersection to ordinary intersection is dfiin2 3979. Theorem intiin 3999 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 3945	. 2 class ∩ 𝑥 ∈ 𝐴 𝐵
5		vy	. . . . . 6 setvar 𝑦
6	5	cv 1374	. . . . 5 class 𝑦
7	6, 3	wcel 2180	. . . 4 wff 𝑦 ∈ 𝐵
8	7, 1, 2	wral 2488	. . 3 wff ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵
9	8, 5	cab 2195	. 2 class {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}
10	4, 9	wceq 1375	1 wff ∩ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}

This definition is referenced by:  eliin  3949  iineq1  3958  iineq2  3961  nfiinxy  3971  nfiinya  3973  nfii1  3975  dfiin2g  3977  cbviin  3982  intiin  3999  0iin  4003  viin  4004  iinxsng  4018  iinxprg  4019  iinuniss  4027  bdciin  16152