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Definition df-iun 3805
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 3835. Theorem uniiun 3853 provides a definition of ordinary union in terms of indexed union. Theorems fniunfv 5625 and funiunfv 5626 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 3803 . 2  class  U_ x  e.  A  B
5 vy . . . . . 6  set  y
65cv 1618 . . . . 5  class  y
76, 3wcel 1621 . . . 4  wff  y  e.  B
87, 1, 2wrex 2510 . . 3  wff  E. x  e.  A  y  e.  B
98, 5cab 2239 . 2  class  { y  |  E. x  e.  A  y  e.  B }
104, 9wceq 1619 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  3807  nfiun  3829  nfiu1  3831  cbviun  3837  iunss  3841  uniiun  3853  iunopab  4189  opeliunxp  4647  reliun  4713  fnasrn  5554  abrexex2g  5620  abrexex2  5632  marypha2lem4  7075  prismorcsetlem  25078  prismorcset  25080  dfiunv2  25082  bnj956  27594  bnj1143  27608  bnj1146  27609  bnj1400  27654  bnj882  27744  bnj18eq1  27745  bnj893  27746  bnj1398  27850
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