ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  df-iun Unicode version

Definition df-iun 3785
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 disjoint variable group (meaning  A cannot depend on  x) and that  B and  x do not share a disjoint 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 3817. Theorem uniiun 3836 provides a definition of ordinary union in terms of indexed union. (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  setvar  x
2 cA . . 3  class  A
3 cB . . 3  class  B
41, 2, 3ciun 3783 . 2  class  U_ x  e.  A  B
5 vy . . . . . 6  setvar  y
65cv 1315 . . . . 5  class  y
76, 3wcel 1465 . . . 4  wff  y  e.  B
87, 1, 2wrex 2394 . . 3  wff  E. x  e.  A  y  e.  B
98, 5cab 2103 . 2  class  { y  |  E. x  e.  A  y  e.  B }
104, 9wceq 1316 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  3787  nfiunxy  3809  nfiunya  3811  nfiu1  3813  dfiunv2  3819  cbviun  3820  iunss  3824  uniiun  3836  iunopab  4173  opeliunxp  4564  reliun  4630  fnasrn  5566  fnasrng  5568  abrexex2g  5986  abrexex2  5990  bdciun  12972
  Copyright terms: Public domain W3C validator