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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 𝑥𝐴 𝐵 = {𝑦 ∣ ∀𝑥𝐴 𝑦𝐵}
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴   𝑦,𝐵
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Detailed syntax breakdown of Definition df-iin
StepHypRef Expression
1 vx . . 3 setvar 𝑥
2 cA . . 3 class 𝐴
3 cB . . 3 class 𝐵
41, 2, 3ciin 4456 . 2 class 𝑥𝐴 𝐵
5 vy . . . . . 6 setvar 𝑦
65cv 1474 . . . . 5 class 𝑦
76, 3wcel 1977 . . . 4 wff 𝑦𝐵
87, 1, 2wral 2896 . . 3 wff 𝑥𝐴 𝑦𝐵
98, 5cab 2596 . 2 class {𝑦 ∣ ∀𝑥𝐴 𝑦𝐵}
104, 9wceq 1475 1 wff 𝑥𝐴 𝐵 = {𝑦 ∣ ∀𝑥𝐴 𝑦𝐵}
 Colors of variables: wff setvar class 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
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