Definition df-vtxdg 16211

Description: Define the vertex degree function for a graph. To be appropriate for arbitrary hypergraphs, we have to double-count those edges that contain 𝑢 "twice" (i.e. self-loops), this being represented as a singleton as the edge's value. Since the degree of a vertex can be (positive) infinity (if the graph containing the vertex is infinite), the extended addition +_𝑒 is used for the summation of the number of "ordinary" edges" and the number of "loops".

Because we cannot in general show that an arbitrary set is either finite or infinite (see inffiexmid 7141), this definition is not as general as it may appear. But we keep it for consistency with the Metamath Proof Explorer. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 20-Dec-2017.) (Revised by AV, 9-Dec-2020.)

Assertion

Ref	Expression
df-vtxdg	⊢ VtxDeg = (𝑔 ∈ V ↦ ⦋(Vtx‘𝑔) / 𝑣⦌⦋(iEdg‘𝑔) / 𝑒⦌(𝑢 ∈ 𝑣 ↦ ((♯‘{𝑥 ∈ dom 𝑒 ∣ 𝑢 ∈ (𝑒‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) = {𝑢}}))))

Distinct variable group:   𝑒,𝑔,𝑢,𝑣,𝑥

Detailed syntax breakdown of Definition df-vtxdg

Step	Hyp	Ref	Expression
1		cvtxdg 16210	. 2 class VtxDeg
2		vg	. . 3 setvar 𝑔
3		cvv 2803	. . 3 class V
4		vv	. . . 4 setvar 𝑣
5	2	cv 1397	. . . . 5 class 𝑔
6		cvtx 15936	. . . . 5 class Vtx
7	5, 6	cfv 5333	. . . 4 class (Vtx‘𝑔)
8		ve	. . . . 5 setvar 𝑒
9		ciedg 15937	. . . . . 6 class iEdg
10	5, 9	cfv 5333	. . . . 5 class (iEdg‘𝑔)
11		vu	. . . . . 6 setvar 𝑢
12	4	cv 1397	. . . . . 6 class 𝑣
13	11	cv 1397	. . . . . . . . . 10 class 𝑢
14		vx	. . . . . . . . . . . 12 setvar 𝑥
15	14	cv 1397	. . . . . . . . . . 11 class 𝑥
16	8	cv 1397	. . . . . . . . . . 11 class 𝑒
17	15, 16	cfv 5333	. . . . . . . . . 10 class (𝑒‘𝑥)
18	13, 17	wcel 2202	. . . . . . . . 9 wff 𝑢 ∈ (𝑒‘𝑥)
19	16	cdm 4731	. . . . . . . . 9 class dom 𝑒
20	18, 14, 19	crab 2515	. . . . . . . 8 class {𝑥 ∈ dom 𝑒 ∣ 𝑢 ∈ (𝑒‘𝑥)}
21		chash 11083	. . . . . . . 8 class ♯
22	20, 21	cfv 5333	. . . . . . 7 class (♯‘{𝑥 ∈ dom 𝑒 ∣ 𝑢 ∈ (𝑒‘𝑥)})
23	13	csn 3673	. . . . . . . . . 10 class {𝑢}
24	17, 23	wceq 1398	. . . . . . . . 9 wff (𝑒‘𝑥) = {𝑢}
25	24, 14, 19	crab 2515	. . . . . . . 8 class {𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) = {𝑢}}
26	25, 21	cfv 5333	. . . . . . 7 class (♯‘{𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) = {𝑢}})
27		cxad 10049	. . . . . . 7 class +𝑒
28	22, 26, 27	co 6028	. . . . . 6 class ((♯‘{𝑥 ∈ dom 𝑒 ∣ 𝑢 ∈ (𝑒‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) = {𝑢}}))
29	11, 12, 28	cmpt 4155	. . . . 5 class (𝑢 ∈ 𝑣 ↦ ((♯‘{𝑥 ∈ dom 𝑒 ∣ 𝑢 ∈ (𝑒‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) = {𝑢}})))
30	8, 10, 29	csb 3128	. . . 4 class ⦋(iEdg‘𝑔) / 𝑒⦌(𝑢 ∈ 𝑣 ↦ ((♯‘{𝑥 ∈ dom 𝑒 ∣ 𝑢 ∈ (𝑒‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) = {𝑢}})))
31	4, 7, 30	csb 3128	. . 3 class ⦋(Vtx‘𝑔) / 𝑣⦌⦋(iEdg‘𝑔) / 𝑒⦌(𝑢 ∈ 𝑣 ↦ ((♯‘{𝑥 ∈ dom 𝑒 ∣ 𝑢 ∈ (𝑒‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) = {𝑢}})))
32	2, 3, 31	cmpt 4155	. 2 class (𝑔 ∈ V ↦ ⦋(Vtx‘𝑔) / 𝑣⦌⦋(iEdg‘𝑔) / 𝑒⦌(𝑢 ∈ 𝑣 ↦ ((♯‘{𝑥 ∈ dom 𝑒 ∣ 𝑢 ∈ (𝑒‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) = {𝑢}}))))
33	1, 32	wceq 1398	1 wff VtxDeg = (𝑔 ∈ V ↦ ⦋(Vtx‘𝑔) / 𝑣⦌⦋(iEdg‘𝑔) / 𝑒⦌(𝑢 ∈ 𝑣 ↦ ((♯‘{𝑥 ∈ dom 𝑒 ∣ 𝑢 ∈ (𝑒‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) = {𝑢}}))))

This definition is referenced by:  vtxdgfval  16212