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Definition df-iun 4095
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 4125. Theorem uniiun 4144 provides a definition of ordinary union in terms of indexed union. Theorems fniunfv 5994 and funiunfv 5995 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 4093 . 2  class  U_ x  e.  A  B
5 vy . . . . . 6  set  y
65cv 1651 . . . . 5  class  y
76, 3wcel 1725 . . . 4  wff  y  e.  B
87, 1, 2wrex 2706 . . 3  wff  E. x  e.  A  y  e.  B
98, 5cab 2422 . 2  class  { y  |  E. x  e.  A  y  e.  B }
104, 9wceq 1652 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  4097  nfiun  4119  nfiu1  4121  dfiunv2  4127  cbviun  4128  iunss  4132  uniiun  4144  iunopab  4486  opeliunxp  4929  reliun  4995  fnasrn  5912  abrexex2g  5988  abrexex2  6001  marypha2lem4  7443  iuneq12daf  24007  iuneq12df  24008  iunrdx  24014  volsupnfl  26251  cshwsiun  28286  bnj956  29147  bnj1143  29161  bnj1146  29162  bnj1400  29207  bnj882  29297  bnj18eq1  29298  bnj893  29299  bnj1398  29403
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