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Definition df-iun 4097
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 4127. Theorem uniiun 4146 provides a definition of ordinary union in terms of indexed union. Theorems fniunfv 5997 and funiunfv 5998 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  set  x
2 cA . . 3  class  A
3 cB . . 3  class  B
41, 2, 3ciun 4095 . 2  class  U_ x  e.  A  B
5 vy . . . . . 6  set  y
65cv 1652 . . . . 5  class  y
76, 3wcel 1726 . . . 4  wff  y  e.  B
87, 1, 2wrex 2708 . . 3  wff  E. x  e.  A  y  e.  B
98, 5cab 2424 . 2  class  { y  |  E. x  e.  A  y  e.  B }
104, 9wceq 1653 1  wff  U_ x  e.  A  B  =  { y  |  E. x  e.  A  y  e.  B }
Colors of variables: wff set class
This definition is referenced by:  eliun  4099  nfiun  4121  nfiu1  4123  dfiunv2  4129  cbviun  4130  iunss  4134  uniiun  4146  iunopab  4489  opeliunxp  4932  reliun  4998  fnasrn  5915  abrexex2g  5991  abrexex2  6004  marypha2lem4  7446  iuneq12daf  24012  iuneq12df  24013  iunrdx  24019  volsupnfl  26262  cshwsiun  28319  bnj956  29220  bnj1143  29234  bnj1146  29235  bnj1400  29280  bnj882  29370  bnj18eq1  29371  bnj893  29372  bnj1398  29476
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