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Definition df-supp 6892
Description: Define the support of a function against a "zero" value. According to Wikipedia ("Support (mathematics)", 31-Mar-2019, https://en.wikipedia.org/wiki/Support_(mathematics)) "In mathematics, the support of a real-valued function f is the subset of the domain containing those elements which are not mapped to zero." and "The notion of support also extends in a natural way to functions taking values in more general sets than R [the real numbers] and to other objects.". The following definition allows for such extensions, being applicable for any sets (which usually are functions) and any element (even not necessarily from the range of the function) regarded as "zero". (Contributed by AV, 31-Mar-2019.) (Revised by AV, 6-Apr-2019.)
Assertion
Ref Expression
df-supp  |- supp  =  ( x  e.  _V , 
z  e.  _V  |->  { i  e.  dom  x  |  ( x " { i } )  =/=  { z } } )
Distinct variable group:    x, i, z

Detailed syntax breakdown of Definition df-supp
StepHypRef Expression
1 csupp 6891 . 2  class supp
2 vx . . 3  setvar  x
3 vz . . 3  setvar  z
4 cvv 3106 . . 3  class  _V
52cv 1397 . . . . . 6  class  x
6 vi . . . . . . . 8  setvar  i
76cv 1397 . . . . . . 7  class  i
87csn 4016 . . . . . 6  class  { i }
95, 8cima 4991 . . . . 5  class  ( x
" { i } )
103cv 1397 . . . . . 6  class  z
1110csn 4016 . . . . 5  class  { z }
129, 11wne 2649 . . . 4  wff  ( x
" { i } )  =/=  { z }
135cdm 4988 . . . 4  class  dom  x
1412, 6, 13crab 2808 . . 3  class  { i  e.  dom  x  |  ( x " {
i } )  =/= 
{ z } }
152, 3, 4, 4, 14cmpt2 6272 . 2  class  ( x  e.  _V ,  z  e.  _V  |->  { i  e.  dom  x  |  ( x " {
i } )  =/= 
{ z } }
)
161, 15wceq 1398 1  wff supp  =  ( x  e.  _V , 
z  e.  _V  |->  { i  e.  dom  x  |  ( x " { i } )  =/=  { z } } )
Colors of variables: wff setvar class
This definition is referenced by:  suppval  6893  supp0prc  6894
  Copyright terms: Public domain W3C validator