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Definition df-iin 3660
Description: Define indexed intersection. Definition of [Stoll] p. 45. See the remarks for its sibling operation of indexed union df-iun 3659. An alternate definition tying indexed intersection to ordinary intersection is dfiin2 3692. Theorem intiin 3711 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 3658 . 2 class 𝑥𝐴 𝐵
5 vy . . . . . 6 setvar 𝑦
65cv 1242 . . . . 5 class 𝑦
76, 3wcel 1393 . . . 4 wff 𝑦𝐵
87, 1, 2wral 2306 . . 3 wff 𝑥𝐴 𝑦𝐵
98, 5cab 2026 . 2 class {𝑦 ∣ ∀𝑥𝐴 𝑦𝐵}
104, 9wceq 1243 1 wff 𝑥𝐴 𝐵 = {𝑦 ∣ ∀𝑥𝐴 𝑦𝐵}
Colors of variables: wff set class
This definition is referenced by:  eliin  3662  iineq1  3671  iineq2  3674  nfiinxy  3684  nfiinya  3686  nfii1  3688  dfiin2g  3690  cbviin  3695  intiin  3711  0iin  3715  viin  3716  iinxsng  3730  iinxprg  3731  iinuniss  3737  bdciin  9973
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