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Definition df-iun 3910
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 3940. Theorem uniiun 3958 provides a definition of ordinary union in terms of indexed union. Theorems fniunfv 5736 and funiunfv 5737 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 groups:    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 3908 . 2  class  U_ x  e.  A  B
5 vy . . . . . 6  set  y
65cv 1624 . . . . 5  class  y
76, 3wcel 1687 . . . 4  wff  y  e.  B
87, 1, 2wrex 2547 . . 3  wff  E. x  e.  A  y  e.  B
98, 5cab 2272 . 2  class  { y  |  E. x  e.  A  y  e.  B }
104, 9wceq 1625 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  3912  nfiun  3934  nfiu1  3936  cbviun  3942  iunss  3946  uniiun  3958  iunopab  4297  opeliunxp  4741  reliun  4807  fnasrn  5665  abrexex2g  5731  abrexex2  5743  marypha2lem4  7188  prismorcsetlem  25313  prismorcset  25315  dfiunv2  25317  bnj956  28077  bnj1143  28091  bnj1146  28092  bnj1400  28137  bnj882  28227  bnj18eq1  28228  bnj893  28229  bnj1398  28333
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