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Definition df-supp 6686
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 6685 . 2  class supp
2 vx . . 3  setvar  x
3 vz . . 3  setvar  z
4 cvv 2966 . . 3  class  _V
52cv 1368 . . . . . 6  class  x
6 vi . . . . . . . 8  setvar  i
76cv 1368 . . . . . . 7  class  i
87csn 3870 . . . . . 6  class  { i }
95, 8cima 4835 . . . . 5  class  ( x
" { i } )
103cv 1368 . . . . . 6  class  z
1110csn 3870 . . . . 5  class  { z }
129, 11wne 2600 . . . 4  wff  ( x
" { i } )  =/=  { z }
135cdm 4832 . . . 4  class  dom  x
1412, 6, 13crab 2713 . . 3  class  { i  e.  dom  x  |  ( x " {
i } )  =/= 
{ z } }
152, 3, 4, 4, 14cmpt2 6088 . 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  6687  supp0prc  6688
  Copyright terms: Public domain W3C validator