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Definition df-supp 6694
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 6693 . 2  class supp
2 vx . . 3  setvar  x
3 vz . . 3  setvar  z
4 cvv 2975 . . 3  class  _V
52cv 1368 . . . . . 6  class  x
6 vi . . . . . . . 8  setvar  i
76cv 1368 . . . . . . 7  class  i
87csn 3880 . . . . . 6  class  { i }
95, 8cima 4846 . . . . 5  class  ( x
" { i } )
103cv 1368 . . . . . 6  class  z
1110csn 3880 . . . . 5  class  { z }
129, 11wne 2609 . . . 4  wff  ( x
" { i } )  =/=  { z }
135cdm 4843 . . . 4  class  dom  x
1412, 6, 13crab 2722 . . 3  class  { i  e.  dom  x  |  ( x " {
i } )  =/= 
{ z } }
152, 3, 4, 4, 14cmpt2 6096 . 2  class  ( x  e.  _V ,  z  e.  _V  |->  { i  e.  dom  x  |  ( x " {
i } )  =/= 
{ z } }
)
161, 15wceq 1369 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  6695  supp0prc  6696
  Copyright terms: Public domain W3C validator