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Definition df-supp 8186
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 8185 . 2 class supp
2 vx . . 3 setvar 𝑥
3 vz . . 3 setvar 𝑧
4 cvv 3480 . . 3 class V
52cv 1539 . . . . . 6 class 𝑥
6 vi . . . . . . . 8 setvar 𝑖
76cv 1539 . . . . . . 7 class 𝑖
87csn 4626 . . . . . 6 class {𝑖}
95, 8cima 5688 . . . . 5 class (𝑥 “ {𝑖})
103cv 1539 . . . . . 6 class 𝑧
1110csn 4626 . . . . 5 class {𝑧}
129, 11wne 2940 . . . 4 wff (𝑥 “ {𝑖}) ≠ {𝑧}
135cdm 5685 . . . 4 class dom 𝑥
1412, 6, 13crab 3436 . . 3 class {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}}
152, 3, 4, 4, 14cmpo 7433 . 2 class (𝑥 ∈ V, 𝑧 ∈ V ↦ {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}})
161, 15wceq 1540 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  8187  supp0prc  8188
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