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Definition df-iun 4304
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 4336. Theorem uniiun 4355 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 4302 . 2  class  U_ x  e.  A  B
5 vy . . . . . 6  setvar  y
65cv 1436 . . . . 5  class  y
76, 3wcel 1870 . . . 4  wff  y  e.  B
87, 1, 2wrex 2783 . . 3  wff  E. x  e.  A  y  e.  B
98, 5cab 2414 . 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  4306  eliun  4307  iuneq12df  4326  nfiun  4330  nfiu1  4332  dfiunv2  4338  cbviun  4339  iunss  4343  uniiun  4355  iunopab  4757  opeliunxp  4906  reliun  4974  fnasrn  6085  abrexex2g  6784  abrexex2  6788  marypha2lem4  7958  cshwsiun  15024  cbviunf  27999  iuneq12daf  28000  iunrdx  28009  bnj956  29367  bnj1143  29381  bnj1146  29382  bnj1400  29426  bnj882  29516  bnj18eq1  29517  bnj893  29518  bnj1398  29622  volsupnfl  31679  ss2iundf  35880  opeliun2xp  38864
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