Definition df-iin 4944

Description: Define indexed intersection. Definition of [Stoll] p. 45. See the remarks for its sibling operation of indexed union df-iun 4943. An alternate definition tying indexed intersection to ordinary intersection is dfiin2 4983. Theorem intiin 5010 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 4942	. 2 class ∩ 𝑥 ∈ 𝐴 𝐵
5		vy	. . . . . 6 setvar 𝑦
6	5	cv 1540	. . . . 5 class 𝑦
7	6, 3	wcel 2113	. . . 4 wff 𝑦 ∈ 𝐵
8	7, 1, 2	wral 3048	. . 3 wff ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵
9	8, 5	cab 2711	. 2 class {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}
10	4, 9	wceq 1541	1 wff ∩ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}

This definition is referenced by:  eliin  4946  iineq1  4959  iineq2  4962  nfiin  4974  nfiing  4976  nfii1  4979  dfiin2g  4981  cbviin  4986  cbviing  4988  cbviinv  4990  intiin  5010  0iin  5014  viin  5015  iinxsng  5038  iinxprg  5039  iinuni  5048  iinabrex  32551  iineq1i  36261  iineq12i  36262  cbviinvw2  36298  cbviindavw  36328  cbviindavw2  36352  iineq12f  38224  iineq12dv  45227