ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  df-iun GIF version

Definition df-iun 3875
Description: Define indexed union. Definition indexed union in [Stoll] p. 45. In most applications, 𝐴 is independent of 𝑥 (although this is not required by the definition), and 𝐵 depends on 𝑥 i.e. can be read informally as 𝐵(𝑥). We call 𝑥 the index, 𝐴 the index set, and 𝐵 the indexed set. In most books, 𝑥𝐴 is written as a subscript or underneath a union symbol . We use a special union symbol to make it easier to distinguish from plain class union. In many theorems, you will see that 𝑥 and 𝐴 are in the same disjoint variable group (meaning 𝐴 cannot depend on 𝑥) and that 𝐵 and 𝑥 do not share a disjoint variable group (meaning that can be thought of as 𝐵(𝑥) i.e. can be substituted with a class expression containing 𝑥). An alternate definition tying indexed union to ordinary union is dfiun2 3907. Theorem uniiun 3926 provides a definition of ordinary union in terms of indexed union. (Contributed by NM, 27-Jun-1998.)
Assertion
Ref Expression
df-iun 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-iun
StepHypRef Expression
1 vx . . 3 setvar 𝑥
2 cA . . 3 class 𝐴
3 cB . . 3 class 𝐵
41, 2, 3ciun 3873 . 2 class 𝑥𝐴 𝐵
5 vy . . . . . 6 setvar 𝑦
65cv 1347 . . . . 5 class 𝑦
76, 3wcel 2141 . . . 4 wff 𝑦𝐵
87, 1, 2wrex 2449 . . 3 wff 𝑥𝐴 𝑦𝐵
98, 5cab 2156 . 2 class {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵}
104, 9wceq 1348 1 wff 𝑥𝐴 𝐵 = {𝑦 ∣ ∃𝑥𝐴 𝑦𝐵}
Colors of variables: wff set class
This definition is referenced by:  eliun  3877  nfiunxy  3899  nfiunya  3901  nfiu1  3903  dfiunv2  3909  cbviun  3910  iunss  3914  uniiun  3926  iunopab  4266  opeliunxp  4666  reliun  4732  fnasrn  5674  fnasrng  5676  abrexex2g  6099  abrexex2  6103  bdciun  13913
  Copyright terms: Public domain W3C validator