Definition df-supp 8178

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 8177	. 2 class supp
2		vx	. . 3 setvar 𝑥
3		vz	. . 3 setvar 𝑧
4		cvv 3450	. . 3 class V
5	2	cv 1568	. . . . . 6 class 𝑥
6		vi	. . . . . . . 8 setvar 𝑖
7	6	cv 1568	. . . . . . 7 class 𝑖
8	7	csn 4583	. . . . . 6 class {𝑖}
9	5, 8	cima 5653	. . . . 5 class (𝑥 “ {𝑖})
10	3	cv 1568	. . . . . 6 class 𝑧
11	10	csn 4583	. . . . 5 class {𝑧}
12	9, 11	wne 2955	. . . 4 wff (𝑥 “ {𝑖}) ≠ {𝑧}
13	5	cdm 5650	. . . 4 class dom 𝑥
14	12, 6, 13	crab 3412	. . 3 class {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}}
15	2, 3, 4, 4, 14	cmpo 7422	. 2 class (𝑥 ∈ V, 𝑧 ∈ V ↦ {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}})
16	1, 15	wceq 1569	1 wff supp = (𝑥 ∈ V, 𝑧 ∈ V ↦ {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}})

This definition is used by:  suppval  8179  supp0prc  8180