Definition df-vtxdg 16299

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 7168), 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 16298	. 2 class VtxDeg
2		vg	. . 3 setvar 𝑔
3		cvv 2815	. . 3 class V
4		vv	. . . 4 setvar 𝑣
5	2	cv 1397	. . . . 5 class 𝑔
6		cvtx 16024	. . . . 5 class Vtx
7	5, 6	cfv 5354	. . . 4 class (Vtx‘𝑔)
8		ve	. . . . 5 setvar 𝑒
9		ciedg 16025	. . . . . 6 class iEdg
10	5, 9	cfv 5354	. . . . 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 5354	. . . . . . . . . 10 class (𝑒‘𝑥)
18	13, 17	wcel 2205	. . . . . . . . 9 wff 𝑢 ∈ (𝑒‘𝑥)
19	16	cdm 4751	. . . . . . . . 9 class dom 𝑒
20	18, 14, 19	crab 2526	. . . . . . . 8 class {𝑥 ∈ dom 𝑒 ∣ 𝑢 ∈ (𝑒‘𝑥)}
21		chash 11142	. . . . . . . 8 class ♯
22	20, 21	cfv 5354	. . . . . . 7 class (♯‘{𝑥 ∈ dom 𝑒 ∣ 𝑢 ∈ (𝑒‘𝑥)})
23	13	csn 3691	. . . . . . . . . 10 class {𝑢}
24	17, 23	wceq 1398	. . . . . . . . 9 wff (𝑒‘𝑥) = {𝑢}
25	24, 14, 19	crab 2526	. . . . . . . 8 class {𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) = {𝑢}}
26	25, 21	cfv 5354	. . . . . . 7 class (♯‘{𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) = {𝑢}})
27		cxad 10106	. . . . . . 7 class +𝑒
28	22, 26, 27	co 6052	. . . . . 6 class ((♯‘{𝑥 ∈ dom 𝑒 ∣ 𝑢 ∈ (𝑒‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) = {𝑢}}))
29	11, 12, 28	cmpt 4173	. . . . 5 class (𝑢 ∈ 𝑣 ↦ ((♯‘{𝑥 ∈ dom 𝑒 ∣ 𝑢 ∈ (𝑒‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) = {𝑢}})))
30	8, 10, 29	csb 3140	. . . 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 3140	. . 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 4173	. 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  16300