MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-iin Structured version   Visualization version   GIF version

Definition df-iin 5000
Description: Define indexed intersection. Definition of [Stoll] p. 45. See the remarks for its sibling operation of indexed union df-iun 4999. An alternate definition tying indexed intersection to ordinary intersection is dfiin2 5037. Theorem intiin 5062 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 4998 . 2 class 𝑥𝐴 𝐵
5 vy . . . . . 6 setvar 𝑦
65cv 1539 . . . . 5 class 𝑦
76, 3wcel 2105 . . . 4 wff 𝑦𝐵
87, 1, 2wral 3060 . . 3 wff 𝑥𝐴 𝑦𝐵
98, 5cab 2708 . 2 class {𝑦 ∣ ∀𝑥𝐴 𝑦𝐵}
104, 9wceq 1540 1 wff 𝑥𝐴 𝐵 = {𝑦 ∣ ∀𝑥𝐴 𝑦𝐵}
Colors of variables: wff setvar class
This definition is referenced by:  eliin  5002  iineq1  5014  iineq2  5017  nfiin  5028  nfiing  5030  nfii1  5032  dfiin2g  5035  cbviin  5040  cbviing  5042  intiin  5062  0iin  5067  viin  5068  iinxsng  5091  iinxprg  5092  iinuni  5101  iinabrex  32068  iineq12f  37336
  Copyright terms: Public domain W3C validator