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Definition df-iun 4087
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 4117. Theorem uniiun 4136 provides a definition of ordinary union in terms of indexed union. Theorems fniunfv 5986 and funiunfv 5987 are useful when  B is a function. (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  set  x
2 cA . . 3  class  A
3 cB . . 3  class  B
41, 2, 3ciun 4085 . 2  class  U_ x  e.  A  B
5 vy . . . . . 6  set  y
65cv 1651 . . . . 5  class  y
76, 3wcel 1725 . . . 4  wff  y  e.  B
87, 1, 2wrex 2698 . . 3  wff  E. x  e.  A  y  e.  B
98, 5cab 2421 . 2  class  { y  |  E. x  e.  A  y  e.  B }
104, 9wceq 1652 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  4089  nfiun  4111  nfiu1  4113  dfiunv2  4119  cbviun  4120  iunss  4124  uniiun  4136  iunopab  4478  opeliunxp  4921  reliun  4987  fnasrn  5904  abrexex2g  5980  abrexex2  5993  marypha2lem4  7435  iuneq12daf  23999  iuneq12df  24000  iunrdx  24006  volsupnfl  26241  cshwsiun  28239  bnj956  29074  bnj1143  29088  bnj1146  29089  bnj1400  29134  bnj882  29224  bnj18eq1  29225  bnj893  29226  bnj1398  29330
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