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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 5985 and funiunfv 5986 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  4920  reliun  4986  fnasrn  5903  abrexex2g  5979  abrexex2  5992  marypha2lem4  7434  iuneq12daf  23995  iuneq12df  23996  iunrdx  24002  volsupnfl  26197  shwrdsiun  28172  bnj956  29001  bnj1143  29015  bnj1146  29016  bnj1400  29061  bnj882  29151  bnj18eq1  29152  bnj893  29153  bnj1398  29257
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