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Definition df-iun 3783
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 3815. Theorem uniiun 3834 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 3781 . 2  class  U_ x  e.  A  B
5 vy . . . . . 6  setvar  y
65cv 1313 . . . . 5  class  y
76, 3wcel 1463 . . . 4  wff  y  e.  B
87, 1, 2wrex 2392 . . 3  wff  E. x  e.  A  y  e.  B
98, 5cab 2101 . 2  class  { y  |  E. x  e.  A  y  e.  B }
104, 9wceq 1314 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  3785  nfiunxy  3807  nfiunya  3809  nfiu1  3811  dfiunv2  3817  cbviun  3818  iunss  3822  uniiun  3834  iunopab  4171  opeliunxp  4562  reliun  4628  fnasrn  5564  fnasrng  5566  abrexex2g  5984  abrexex2  5988  bdciun  12887
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