Definition df-iin 4458

Description: Define indexed intersection. Definition of [Stoll] p. 45. See the remarks for its sibling operation of indexed union df-iun 4457. An alternate definition tying indexed intersection to ordinary intersection is dfiin2 4491. Theorem intiin 4510 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 4456	. 2 class ∩ 𝑥 ∈ 𝐴 𝐵
5		vy	. . . . . 6 setvar 𝑦
6	5	cv 1474	. . . . 5 class 𝑦
7	6, 3	wcel 1977	. . . 4 wff 𝑦 ∈ 𝐵
8	7, 1, 2	wral 2896	. . 3 wff ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵
9	8, 5	cab 2596	. 2 class {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}
10	4, 9	wceq 1475	1 wff ∩ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}

This definition is referenced by:  eliin  4461  iineq1  4471  iineq2  4474  nfiin  4485  nfii1  4487  dfiin2g  4489  cbviin  4494  intiin  4510  0iin  4514  viin  4515  iinxsng  4536  iinxprg  4537  iinuni  4545  iineq12f  33143