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Definition df-iun 3907
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 3937. Theorem uniiun 3955 provides a definition of ordinary union in terms of indexed union. Theorems fniunfv 5773 and funiunfv 5774 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 3905 . 2  class  U_ x  e.  A  B
5 vy . . . . . 6  set  y
65cv 1622 . . . . 5  class  y
76, 3wcel 1684 . . . 4  wff  y  e.  B
87, 1, 2wrex 2544 . . 3  wff  E. x  e.  A  y  e.  B
98, 5cab 2269 . 2  class  { y  |  E. x  e.  A  y  e.  B }
104, 9wceq 1623 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  3909  nfiun  3931  nfiu1  3933  cbviun  3939  iunss  3943  uniiun  3955  iunopab  4296  opeliunxp  4740  reliun  4806  fnasrn  5702  abrexex2g  5768  abrexex2  5780  marypha2lem4  7191  iuneq12daf  23154  iuneq12df  23155  ssiun3  23156  iunrdx  23161  prismorcsetlem  25912  prismorcset  25914  dfiunv2  25916  bnj956  28808  bnj1143  28822  bnj1146  28823  bnj1400  28868  bnj882  28958  bnj18eq1  28959  bnj893  28960  bnj1398  29064
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