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Definition df-iun 4250
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 4282. Theorem uniiun 4301 provides a definition of ordinary union in terms of indexed union. Theorems fniunfv 6138 and funiunfv 6139 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  setvar  x
2 cA . . 3  class  A
3 cB . . 3  class  B
41, 2, 3ciun 4248 . 2  class  U_ x  e.  A  B
5 vy . . . . . 6  setvar  y
65cv 1447 . . . . 5  class  y
76, 3wcel 1891 . . . 4  wff  y  e.  B
87, 1, 2wrex 2738 . . 3  wff  E. x  e.  A  y  e.  B
98, 5cab 2438 . 2  class  { y  |  E. x  e.  A  y  e.  B }
104, 9wceq 1448 1  wff  U_ x  e.  A  B  =  { y  |  E. x  e.  A  y  e.  B }
Colors of variables: wff setvar class
This definition is referenced by:  rabasiun  4252  eliun  4253  iuneq12df  4272  nfiun  4276  nfiu1  4278  dfiunv2  4284  cbviun  4285  iunss  4289  uniiun  4301  iunopab  4710  opeliunxp  4864  reliun  4932  fnasrn  6055  abrexex2g  6758  abrexex2  6762  marypha2lem4  7939  cshwsiun  15081  cbviunf  28180  iuneq12daf  28181  iunrdx  28189  bnj956  29594  bnj1143  29608  bnj1146  29609  bnj1400  29653  bnj882  29743  bnj18eq1  29744  bnj893  29745  bnj1398  29849  volsupnfl  31987  ss2iundf  36253  iunssf  37437  opeliun2xp  40439
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