Definition df-supp 8162

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 8161	. 2 class supp
2		vx	. . 3 setvar 𝑥
3		vz	. . 3 setvar 𝑧
4		cvv 3453	. . 3 class V
5	2	cv 1569	. . . . . 6 class 𝑥
6		vi	. . . . . . . 8 setvar 𝑖
7	6	cv 1569	. . . . . . 7 class 𝑖
8	7	csn 4587	. . . . . 6 class {𝑖}
9	5, 8	cima 5662	. . . . 5 class (𝑥 “ {𝑖})
10	3	cv 1569	. . . . . 6 class 𝑧
11	10	csn 4587	. . . . 5 class {𝑧}
12	9, 11	wne 2957	. . . 4 wff (𝑥 “ {𝑖}) ≠ {𝑧}
13	5	cdm 5659	. . . 4 class dom 𝑥
14	12, 6, 13	crab 3414	. . 3 class {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}}
15	2, 3, 4, 4, 14	cmpo 7418	. 2 class (𝑥 ∈ V, 𝑧 ∈ V ↦ {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}})
16	1, 15	wceq 1570	1 wff supp = (𝑥 ∈ V, 𝑧 ∈ V ↦ {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}})

This definition is used by:  suppval  8163  supp0prc  8164