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Definition df-iun 4301
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 4333. Theorem uniiun 4352 provides a definition of ordinary union in terms of indexed union. Theorems fniunfv 6167 and funiunfv 6168 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 4299 . 2  class  U_ x  e.  A  B
5 vy . . . . . 6  setvar  y
65cv 1436 . . . . 5  class  y
76, 3wcel 1872 . . . 4  wff  y  e.  B
87, 1, 2wrex 2772 . . 3  wff  E. x  e.  A  y  e.  B
98, 5cab 2407 . 2  class  { y  |  E. x  e.  A  y  e.  B }
104, 9wceq 1437 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  4303  eliun  4304  iuneq12df  4323  nfiun  4327  nfiu1  4329  dfiunv2  4335  cbviun  4336  iunss  4340  uniiun  4352  iunopab  4756  opeliunxp  4905  reliun  4973  fnasrn  6085  abrexex2g  6784  abrexex2  6788  marypha2lem4  7961  cshwsiun  15069  cbviunf  28171  iuneq12daf  28172  iunrdx  28181  bnj956  29596  bnj1143  29610  bnj1146  29611  bnj1400  29655  bnj882  29745  bnj18eq1  29746  bnj893  29747  bnj1398  29851  volsupnfl  31949  ss2iundf  36221  opeliun2xp  39736
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