Definition df-supp 8149

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

Step	Hyp	Ref	Expression
1		csupp 8148	. 2 class supp
2		vx	. . 3 setvar 𝑥
3		vz	. . 3 setvar 𝑧
4		cvv 3472	. . 3 class V
5	2	cv 1538	. . . . . 6 class 𝑥
6		vi	. . . . . . . 8 setvar 𝑖
7	6	cv 1538	. . . . . . 7 class 𝑖
8	7	csn 4627	. . . . . 6 class {𝑖}
9	5, 8	cima 5678	. . . . 5 class (𝑥 “ {𝑖})
10	3	cv 1538	. . . . . 6 class 𝑧
11	10	csn 4627	. . . . 5 class {𝑧}
12	9, 11	wne 2938	. . . 4 wff (𝑥 “ {𝑖}) ≠ {𝑧}
13	5	cdm 5675	. . . 4 class dom 𝑥
14	12, 6, 13	crab 3430	. . 3 class {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}}
15	2, 3, 4, 4, 14	cmpo 7413	. 2 class (𝑥 ∈ V, 𝑧 ∈ V ↦ {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}})
16	1, 15	wceq 1539	1 wff supp = (𝑥 ∈ V, 𝑧 ∈ V ↦ {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}})

This definition is referenced by:  suppval  8150  supp0prc  8151