Definition df-iin 3915

Description: Define indexed intersection. Definition of [Stoll] p. 45. See the remarks for its sibling operation of indexed union df-iun 3914. An alternate definition tying indexed intersection to ordinary intersection is dfiin2 3947. Theorem intiin 3967 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 3913	. 2 class ∩ 𝑥 ∈ 𝐴 𝐵
5		vy	. . . . . 6 setvar 𝑦
6	5	cv 1363	. . . . 5 class 𝑦
7	6, 3	wcel 2164	. . . 4 wff 𝑦 ∈ 𝐵
8	7, 1, 2	wral 2472	. . 3 wff ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵
9	8, 5	cab 2179	. 2 class {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}
10	4, 9	wceq 1364	1 wff ∩ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}

This definition is referenced by:  eliin  3917  iineq1  3926  iineq2  3929  nfiinxy  3939  nfiinya  3941  nfii1  3943  dfiin2g  3945  cbviin  3950  intiin  3967  0iin  3971  viin  3972  iinxsng  3986  iinxprg  3987  iinuniss  3995  bdciin  15371