Definition df-supp 8147

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 8146	. 2 class supp
2		vx	. . 3 setvar 𝑥
3		vz	. . 3 setvar 𝑧
4		cvv 3475	. . 3 class V
5	2	cv 1541	. . . . . 6 class 𝑥
6		vi	. . . . . . . 8 setvar 𝑖
7	6	cv 1541	. . . . . . 7 class 𝑖
8	7	csn 4629	. . . . . 6 class {𝑖}
9	5, 8	cima 5680	. . . . 5 class (𝑥 “ {𝑖})
10	3	cv 1541	. . . . . 6 class 𝑧
11	10	csn 4629	. . . . 5 class {𝑧}
12	9, 11	wne 2941	. . . 4 wff (𝑥 “ {𝑖}) ≠ {𝑧}
13	5	cdm 5677	. . . 4 class dom 𝑥
14	12, 6, 13	crab 3433	. . 3 class {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}}
15	2, 3, 4, 4, 14	cmpo 7411	. 2 class (𝑥 ∈ V, 𝑧 ∈ V ↦ {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}})
16	1, 15	wceq 1542	1 wff supp = (𝑥 ∈ V, 𝑧 ∈ V ↦ {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}})

This definition is referenced by:  suppval  8148  supp0prc  8149