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Definition df-iun 3888
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 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 3920. Theorem uniiun 3940 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 3886 . 2  class  U_ x  e.  A  B
5 vy . . . . . 6  setvar  y
65cv 1352 . . . . 5  class  y
76, 3wcel 2148 . . . 4  wff  y  e.  B
87, 1, 2wrex 2456 . . 3  wff  E. x  e.  A  y  e.  B
98, 5cab 2163 . 2  class  { y  |  E. x  e.  A  y  e.  B }
104, 9wceq 1353 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  3890  nfiunxy  3912  nfiunya  3914  nfiu1  3916  dfiunv2  3922  cbviun  3923  iunss  3927  uniiun  3940  iunopab  4281  opeliunxp  4681  reliun  4747  fnasrn  5694  fnasrng  5696  abrexex2g  6120  abrexex2  6124  bdciun  14600
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