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Definition df-iun 4245
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 4277. Theorem uniiun 4296 provides a definition of ordinary union in terms of indexed union. Theorems fniunfv 6060 and funiunfv 6061 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 4243 . 2  class  U_ x  e.  A  B
5 vy . . . . . 6  setvar  y
65cv 1398 . . . . 5  class  y
76, 3wcel 1826 . . . 4  wff  y  e.  B
87, 1, 2wrex 2733 . . 3  wff  E. x  e.  A  y  e.  B
98, 5cab 2367 . 2  class  { y  |  E. x  e.  A  y  e.  B }
104, 9wceq 1399 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  4247  eliun  4248  iuneq12df  4267  nfiun  4271  nfiu1  4273  dfiunv2  4279  cbviun  4280  iunss  4284  uniiun  4296  iunopab  4697  opeliunxp  4965  reliun  5035  fnasrn  5979  abrexex2g  6676  abrexex2  6680  marypha2lem4  7813  cshwsiun  14586  cbviunf  27550  iuneq12daf  27551  iunrdx  27560  volsupnfl  30224  opeliun2xp  33122  bnj956  34182  bnj1143  34196  bnj1146  34197  bnj1400  34241  bnj882  34331  bnj18eq1  34332  bnj893  34333  bnj1398  34437
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