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Definition df-iun 4322
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 4354. Theorem uniiun 4373 provides a definition of ordinary union in terms of indexed union. Theorems fniunfv 6140 and funiunfv 6141 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 4320 . 2  class  U_ x  e.  A  B
5 vy . . . . . 6  setvar  y
65cv 1373 . . . . 5  class  y
76, 3wcel 1762 . . . 4  wff  y  e.  B
87, 1, 2wrex 2810 . . 3  wff  E. x  e.  A  y  e.  B
98, 5cab 2447 . 2  class  { y  |  E. x  e.  A  y  e.  B }
104, 9wceq 1374 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  4324  eliun  4325  iuneq12df  4344  nfiun  4348  nfiu1  4350  dfiunv2  4356  cbviun  4357  iunss  4361  uniiun  4373  iunopab  4778  opeliunxp  5045  reliun  5116  fnasrn  6060  abrexex2g  6753  abrexex2  6757  marypha2lem4  7889  cshwsiun  14433  iuneq12daf  27086  iunrdx  27092  volsupnfl  29625  opeliun2xp  31863  bnj956  32791  bnj1143  32805  bnj1146  32806  bnj1400  32850  bnj882  32940  bnj18eq1  32941  bnj893  32942  bnj1398  33046
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