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Definition df-iin 4555
Description: Define indexed intersection. Definition of [Stoll] p. 45. See the remarks for its sibling operation of indexed union df-iun 4554. An alternate definition tying indexed intersection to ordinary intersection is dfiin2 4587. Theorem intiin 4606 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 4553 . 2 class 𝑥𝐴 𝐵
5 vy . . . . . 6 setvar 𝑦
65cv 1522 . . . . 5 class 𝑦
76, 3wcel 2030 . . . 4 wff 𝑦𝐵
87, 1, 2wral 2941 . . 3 wff 𝑥𝐴 𝑦𝐵
98, 5cab 2637 . 2 class {𝑦 ∣ ∀𝑥𝐴 𝑦𝐵}
104, 9wceq 1523 1 wff 𝑥𝐴 𝐵 = {𝑦 ∣ ∀𝑥𝐴 𝑦𝐵}
Colors of variables: wff setvar class
This definition is referenced by:  eliin  4557  iineq1  4567  iineq2  4570  nfiin  4581  nfii1  4583  dfiin2g  4585  cbviin  4590  intiin  4606  0iin  4610  viin  4611  iinxsng  4632  iinxprg  4633  iinuni  4641  iineq12f  34103
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