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Definition df-supp 6863
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 6862 . 2  class supp
2 vx . . 3  setvar  x
3 vz . . 3  setvar  z
4 cvv 3016 . . 3  class  _V
52cv 1436 . . . . . 6  class  x
6 vi . . . . . . . 8  setvar  i
76cv 1436 . . . . . . 7  class  i
87csn 3934 . . . . . 6  class  { i }
95, 8cima 4792 . . . . 5  class  ( x
" { i } )
103cv 1436 . . . . . 6  class  z
1110csn 3934 . . . . 5  class  { z }
129, 11wne 2593 . . . 4  wff  ( x
" { i } )  =/=  { z }
135cdm 4789 . . . 4  class  dom  x
1412, 6, 13crab 2712 . . 3  class  { i  e.  dom  x  |  ( x " {
i } )  =/= 
{ z } }
152, 3, 4, 4, 14cmpt2 6244 . 2  class  ( x  e.  _V ,  z  e.  _V  |->  { i  e.  dom  x  |  ( x " {
i } )  =/= 
{ z } }
)
161, 15wceq 1437 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  6864  supp0prc  6865
  Copyright terms: Public domain W3C validator