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Definition df-supp 6903
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 6902 . 2  class supp
2 vx . . 3  setvar  x
3 vz . . 3  setvar  z
4 cvv 3113 . . 3  class  _V
52cv 1378 . . . . . 6  class  x
6 vi . . . . . . . 8  setvar  i
76cv 1378 . . . . . . 7  class  i
87csn 4027 . . . . . 6  class  { i }
95, 8cima 5002 . . . . 5  class  ( x
" { i } )
103cv 1378 . . . . . 6  class  z
1110csn 4027 . . . . 5  class  { z }
129, 11wne 2662 . . . 4  wff  ( x
" { i } )  =/=  { z }
135cdm 4999 . . . 4  class  dom  x
1412, 6, 13crab 2818 . . 3  class  { i  e.  dom  x  |  ( x " {
i } )  =/= 
{ z } }
152, 3, 4, 4, 14cmpt2 6287 . 2  class  ( x  e.  _V ,  z  e.  _V  |->  { i  e.  dom  x  |  ( x " {
i } )  =/= 
{ z } }
)
161, 15wceq 1379 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  6904  supp0prc  6905
  Copyright terms: Public domain W3C validator