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Definition df-iin 4452
Description: Define indexed intersection. Definition of [Stoll] p. 45. See the remarks for its sibling operation of indexed union df-iun 4451. An alternate definition tying indexed intersection to ordinary intersection is dfiin2 4485. Theorem intiin 4504 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 4450 . 2 class 𝑥𝐴 𝐵
5 vy . . . . . 6 setvar 𝑦
65cv 1473 . . . . 5 class 𝑦
76, 3wcel 1976 . . . 4 wff 𝑦𝐵
87, 1, 2wral 2895 . . 3 wff 𝑥𝐴 𝑦𝐵
98, 5cab 2595 . 2 class {𝑦 ∣ ∀𝑥𝐴 𝑦𝐵}
104, 9wceq 1474 1 wff 𝑥𝐴 𝐵 = {𝑦 ∣ ∀𝑥𝐴 𝑦𝐵}
Colors of variables: wff setvar class
This definition is referenced by:  eliin  4455  iineq1  4465  iineq2  4468  nfiin  4479  nfii1  4481  dfiin2g  4483  cbviin  4488  intiin  4504  0iin  4508  viin  4509  iinxsng  4530  iinxprg  4531  iinuni  4539  iineq12f  32926
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