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Definition df-iun 3923
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 3953. Theorem uniiun 3971 provides a definition of ordinary union in terms of indexed union. Theorems fniunfv 5789 and funiunfv 5790 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 3921 . 2  class  U_ x  e.  A  B
5 vy . . . . . 6  set  y
65cv 1631 . . . . 5  class  y
76, 3wcel 1696 . . . 4  wff  y  e.  B
87, 1, 2wrex 2557 . . 3  wff  E. x  e.  A  y  e.  B
98, 5cab 2282 . 2  class  { y  |  E. x  e.  A  y  e.  B }
104, 9wceq 1632 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  3925  nfiun  3947  nfiu1  3949  cbviun  3955  iunss  3959  uniiun  3971  iunopab  4312  opeliunxp  4756  reliun  4822  fnasrn  5718  abrexex2g  5784  abrexex2  5796  marypha2lem4  7207  iuneq12daf  23170  iuneq12df  23171  ssiun3  23172  iunrdx  23177  prismorcsetlem  26015  prismorcset  26017  dfiunv2  26019  bnj956  29124  bnj1143  29138  bnj1146  29139  bnj1400  29184  bnj882  29274  bnj18eq1  29275  bnj893  29276  bnj1398  29380
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