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Mirrors > Home > MPE Home > Th. List > df-supp | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
df-supp | ⊢ supp = (𝑥 ∈ V, 𝑧 ∈ V ↦ {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csupp 7182 | . 2 class supp | |
2 | vx | . . 3 setvar 𝑥 | |
3 | vz | . . 3 setvar 𝑧 | |
4 | cvv 3173 | . . 3 class V | |
5 | 2 | cv 1474 | . . . . . 6 class 𝑥 |
6 | vi | . . . . . . . 8 setvar 𝑖 | |
7 | 6 | cv 1474 | . . . . . . 7 class 𝑖 |
8 | 7 | csn 4125 | . . . . . 6 class {𝑖} |
9 | 5, 8 | cima 5041 | . . . . 5 class (𝑥 “ {𝑖}) |
10 | 3 | cv 1474 | . . . . . 6 class 𝑧 |
11 | 10 | csn 4125 | . . . . 5 class {𝑧} |
12 | 9, 11 | wne 2780 | . . . 4 wff (𝑥 “ {𝑖}) ≠ {𝑧} |
13 | 5 | cdm 5038 | . . . 4 class dom 𝑥 |
14 | 12, 6, 13 | crab 2900 | . . 3 class {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}} |
15 | 2, 3, 4, 4, 14 | cmpt2 6551 | . 2 class (𝑥 ∈ V, 𝑧 ∈ V ↦ {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}}) |
16 | 1, 15 | wceq 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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