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Definition df-iin 4998
Description: Define indexed intersection. Definition of [Stoll] p. 45. See the remarks for its sibling operation of indexed union df-iun 4997. An alternate definition tying indexed intersection to ordinary intersection is dfiin2 5038. Theorem intiin 5063 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 4996 . 2 class 𝑥𝐴 𝐵
5 vy . . . . . 6 setvar 𝑦
65cv 1535 . . . . 5 class 𝑦
76, 3wcel 2105 . . . 4 wff 𝑦𝐵
87, 1, 2wral 3058 . . 3 wff 𝑥𝐴 𝑦𝐵
98, 5cab 2711 . 2 class {𝑦 ∣ ∀𝑥𝐴 𝑦𝐵}
104, 9wceq 1536 1 wff 𝑥𝐴 𝐵 = {𝑦 ∣ ∀𝑥𝐴 𝑦𝐵}
Colors of variables: wff setvar class
This definition is referenced by:  eliin  5000  iineq1  5013  iineq2  5016  nfiin  5028  nfiing  5030  nfii1  5033  dfiin2g  5036  cbviin  5041  cbviing  5043  cbviinv  5045  intiin  5063  0iin  5068  viin  5069  iinxsng  5092  iinxprg  5093  iinuni  5102  iinabrex  32588  iineq1i  36177  iineq12i  36178  cbviinvw2  36215  cbviindavw  36245  cbviindavw2  36269  iineq12f  38150  iineq12dv  45045
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