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Definition df-supp 7341
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 ∈ V, 𝑧 ∈ V ↦ {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}})
Distinct variable group:   𝑥,𝑖,𝑧

Detailed syntax breakdown of Definition df-supp
StepHypRef Expression
1 csupp 7340 . 2 class supp
2 vx . . 3 setvar 𝑥
3 vz . . 3 setvar 𝑧
4 cvv 3231 . . 3 class V
52cv 1522 . . . . . 6 class 𝑥
6 vi . . . . . . . 8 setvar 𝑖
76cv 1522 . . . . . . 7 class 𝑖
87csn 4210 . . . . . 6 class {𝑖}
95, 8cima 5146 . . . . 5 class (𝑥 “ {𝑖})
103cv 1522 . . . . . 6 class 𝑧
1110csn 4210 . . . . 5 class {𝑧}
129, 11wne 2823 . . . 4 wff (𝑥 “ {𝑖}) ≠ {𝑧}
135cdm 5143 . . . 4 class dom 𝑥
1412, 6, 13crab 2945 . . 3 class {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}}
152, 3, 4, 4, 14cmpt2 6692 . 2 class (𝑥 ∈ V, 𝑧 ∈ V ↦ {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}})
161, 15wceq 1523 1 wff supp = (𝑥 ∈ V, 𝑧 ∈ V ↦ {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}})
Colors of variables: wff setvar class
This definition is referenced by:  suppval  7342  supp0prc  7343
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