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Definition df-supp 7183
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 7182 . 2 class supp
2 vx . . 3 setvar 𝑥
3 vz . . 3 setvar 𝑧
4 cvv 3173 . . 3 class V
52cv 1474 . . . . . 6 class 𝑥
6 vi . . . . . . . 8 setvar 𝑖
76cv 1474 . . . . . . 7 class 𝑖
87csn 4125 . . . . . 6 class {𝑖}
95, 8cima 5041 . . . . 5 class (𝑥 “ {𝑖})
103cv 1474 . . . . . 6 class 𝑧
1110csn 4125 . . . . 5 class {𝑧}
129, 11wne 2780 . . . 4 wff (𝑥 “ {𝑖}) ≠ {𝑧}
135cdm 5038 . . . 4 class dom 𝑥
1412, 6, 13crab 2900 . . 3 class {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}}
152, 3, 4, 4, 14cmpt2 6551 . 2 class (𝑥 ∈ V, 𝑧 ∈ V ↦ {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}})
161, 15wceq 1475 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  7184  supp0prc  7185
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