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Definition df-iun 4317
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 4349. Theorem uniiun 4368 provides a definition of ordinary union in terms of indexed union. Theorems fniunfv 6144 and funiunfv 6145 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 4315 . 2  class  U_ x  e.  A  B
5 vy . . . . . 6  setvar  y
65cv 1382 . . . . 5  class  y
76, 3wcel 1804 . . . 4  wff  y  e.  B
87, 1, 2wrex 2794 . . 3  wff  E. x  e.  A  y  e.  B
98, 5cab 2428 . 2  class  { y  |  E. x  e.  A  y  e.  B }
104, 9wceq 1383 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  4319  eliun  4320  iuneq12df  4339  nfiun  4343  nfiu1  4345  dfiunv2  4351  cbviun  4352  iunss  4356  uniiun  4368  iunopab  4773  opeliunxp  5041  reliun  5113  fnasrn  6062  abrexex2g  6762  abrexex2  6766  marypha2lem4  7900  cshwsiun  14566  iuneq12daf  27403  iunrdx  27409  volsupnfl  30035  opeliun2xp  32790  bnj956  33703  bnj1143  33717  bnj1146  33718  bnj1400  33762  bnj882  33852  bnj18eq1  33853  bnj893  33854  bnj1398  33958
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