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Definition df-supp 7822
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 7821 . 2 class supp
2 vx . . 3 setvar 𝑥
3 vz . . 3 setvar 𝑧
4 cvv 3495 . . 3 class V
52cv 1527 . . . . . 6 class 𝑥
6 vi . . . . . . . 8 setvar 𝑖
76cv 1527 . . . . . . 7 class 𝑖
87csn 4559 . . . . . 6 class {𝑖}
95, 8cima 5552 . . . . 5 class (𝑥 “ {𝑖})
103cv 1527 . . . . . 6 class 𝑧
1110csn 4559 . . . . 5 class {𝑧}
129, 11wne 3016 . . . 4 wff (𝑥 “ {𝑖}) ≠ {𝑧}
135cdm 5549 . . . 4 class dom 𝑥
1412, 6, 13crab 3142 . . 3 class {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}}
152, 3, 4, 4, 14cmpo 7147 . 2 class (𝑥 ∈ V, 𝑧 ∈ V ↦ {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}})
161, 15wceq 1528 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  7823  supp0prc  7824
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