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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 7821 | . 2 class supp | |
2 | vx | . . 3 setvar 𝑥 | |
3 | vz | . . 3 setvar 𝑧 | |
4 | cvv 3495 | . . 3 class V | |
5 | 2 | cv 1527 | . . . . . 6 class 𝑥 |
6 | vi | . . . . . . . 8 setvar 𝑖 | |
7 | 6 | cv 1527 | . . . . . . 7 class 𝑖 |
8 | 7 | csn 4559 | . . . . . 6 class {𝑖} |
9 | 5, 8 | cima 5552 | . . . . 5 class (𝑥 “ {𝑖}) |
10 | 3 | cv 1527 | . . . . . 6 class 𝑧 |
11 | 10 | csn 4559 | . . . . 5 class {𝑧} |
12 | 9, 11 | wne 3016 | . . . 4 wff (𝑥 “ {𝑖}) ≠ {𝑧} |
13 | 5 | cdm 5549 | . . . 4 class dom 𝑥 |
14 | 12, 6, 13 | crab 3142 | . . 3 class {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}} |
15 | 2, 3, 4, 4, 14 | cmpo 7147 | . 2 class (𝑥 ∈ V, 𝑧 ∈ V ↦ {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}}) |
16 | 1, 15 | wceq 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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