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Definition df-iun 4055
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 4085. Theorem uniiun 4104 provides a definition of ordinary union in terms of indexed union. Theorems fniunfv 5953 and funiunfv 5954 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 4053 . 2  class  U_ x  e.  A  B
5 vy . . . . . 6  set  y
65cv 1648 . . . . 5  class  y
76, 3wcel 1721 . . . 4  wff  y  e.  B
87, 1, 2wrex 2667 . . 3  wff  E. x  e.  A  y  e.  B
98, 5cab 2390 . 2  class  { y  |  E. x  e.  A  y  e.  B }
104, 9wceq 1649 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  4057  nfiun  4079  nfiu1  4081  dfiunv2  4087  cbviun  4088  iunss  4092  uniiun  4104  iunopab  4446  opeliunxp  4888  reliun  4954  fnasrn  5871  abrexex2g  5947  abrexex2  5960  marypha2lem4  7401  iuneq12daf  23960  iuneq12df  23961  iunrdx  23967  volsupnfl  26150  bnj956  28853  bnj1143  28867  bnj1146  28868  bnj1400  28913  bnj882  29003  bnj18eq1  29004  bnj893  29005  bnj1398  29109
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