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Definition df-iun 3686
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,  x  e.  A is written as a subscript or underneath a union symbol  U.. We use a special union symbol  U_ to make it easier to distinguish from plain class union. In many theorems, you will see that  x and 
A are in the same distinct variable group (meaning  A cannot depend on  x) and that  B and  x do not share a distinct 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 3718. Theorem uniiun 3737 provides a definition of ordinary union in terms of indexed union. (Contributed by NM, 27-Jun-1998.)
Assertion
Ref Expression
df-iun  |-  U_ x  e.  A  B  =  { y  |  E. x  e.  A  y  e.  B }
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 3684 . 2  class  U_ x  e.  A  B
5 vy . . . . . 6  setvar  y
65cv 1258 . . . . 5  class  y
76, 3wcel 1409 . . . 4  wff  y  e.  B
87, 1, 2wrex 2324 . . 3  wff  E. x  e.  A  y  e.  B
98, 5cab 2042 . 2  class  { y  |  E. x  e.  A  y  e.  B }
104, 9wceq 1259 1  wff  U_ x  e.  A  B  =  { y  |  E. x  e.  A  y  e.  B }
Colors of variables: wff set class
This definition is referenced by:  eliun  3688  nfiunxy  3710  nfiunya  3712  nfiu1  3714  dfiunv2  3720  cbviun  3721  iunss  3725  uniiun  3737  iunopab  4045  opeliunxp  4422  reliun  4485  fnasrn  5368  fnasrng  5370  abrexex2g  5774  abrexex2  5778  bdciun  10364
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