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Definition df-iun 3867
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 3897. Theorem uniiun 3915 provides a definition of ordinary union in terms of indexed union. Theorems fniunfv 5693 and funiunfv 5694 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 3865 . 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 2517 . . 3  wff  E. x  e.  A  y  e.  B
98, 5cab 2242 . 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  3869  nfiun  3891  nfiu1  3893  cbviun  3899  iunss  3903  uniiun  3915  iunopab  4254  opeliunxp  4714  reliun  4780  fnasrn  5622  abrexex2g  5688  abrexex2  5700  marypha2lem4  7145  prismorcsetlem  25265  prismorcset  25267  dfiunv2  25269  bnj956  27841  bnj1143  27855  bnj1146  27856  bnj1400  27901  bnj882  27991  bnj18eq1  27992  bnj893  27993  bnj1398  28097
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