Definition df-supp 7831

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 7830	. 2 class supp
2		vx	. . 3 setvar 𝑥
3		vz	. . 3 setvar 𝑧
4		cvv 3494	. . 3 class V
5	2	cv 1536	. . . . . 6 class 𝑥
6		vi	. . . . . . . 8 setvar 𝑖
7	6	cv 1536	. . . . . . 7 class 𝑖
8	7	csn 4567	. . . . . 6 class {𝑖}
9	5, 8	cima 5558	. . . . 5 class (𝑥 “ {𝑖})
10	3	cv 1536	. . . . . 6 class 𝑧
11	10	csn 4567	. . . . 5 class {𝑧}
12	9, 11	wne 3016	. . . 4 wff (𝑥 “ {𝑖}) ≠ {𝑧}
13	5	cdm 5555	. . . 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 7158	. 2 class (𝑥 ∈ V, 𝑧 ∈ V ↦ {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}})
16	1, 15	wceq 1537	1 wff supp = (𝑥 ∈ V, 𝑧 ∈ V ↦ {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}})

This definition is referenced by:  suppval  7832  supp0prc  7833