Definition df-iin 4928

Description: Define indexed intersection. Definition of [Stoll] p. 45. See the remarks for its sibling operation of indexed union df-iun 4927. An alternate definition tying indexed intersection to ordinary intersection is dfiin2 4965. Theorem intiin 4990 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 4926	. 2 class ∩ 𝑥 ∈ 𝐴 𝐵
5		vy	. . . . . 6 setvar 𝑦
6	5	cv 1538	. . . . 5 class 𝑦
7	6, 3	wcel 2107	. . . 4 wff 𝑦 ∈ 𝐵
8	7, 1, 2	wral 3065	. . 3 wff ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵
9	8, 5	cab 2716	. 2 class {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}
10	4, 9	wceq 1539	1 wff ∩ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}

This definition is referenced by:  eliin  4930  iineq1  4942  iineq2  4945  nfiin  4956  nfiing  4958  nfii1  4960  dfiin2g  4963  cbviin  4968  cbviing  4970  intiin  4990  0iin  4994  viin  4995  iinxsng  5018  iinxprg  5019  iinuni  5028  iinabrex  30917  iineq12f  36331