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Definition df-supp 7293
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 7292 . 2 class supp
2 vx . . 3 setvar 𝑥
3 vz . . 3 setvar 𝑧
4 cvv 3198 . . 3 class V
52cv 1481 . . . . . 6 class 𝑥
6 vi . . . . . . . 8 setvar 𝑖
76cv 1481 . . . . . . 7 class 𝑖
87csn 4175 . . . . . 6 class {𝑖}
95, 8cima 5115 . . . . 5 class (𝑥 “ {𝑖})
103cv 1481 . . . . . 6 class 𝑧
1110csn 4175 . . . . 5 class {𝑧}
129, 11wne 2793 . . . 4 wff (𝑥 “ {𝑖}) ≠ {𝑧}
135cdm 5112 . . . 4 class dom 𝑥
1412, 6, 13crab 2915 . . 3 class {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}}
152, 3, 4, 4, 14cmpt2 6649 . 2 class (𝑥 ∈ V, 𝑧 ∈ V ↦ {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}})
161, 15wceq 1482 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  7294  supp0prc  7295
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