Definition df-vtxdg 16046

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 7079), 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 16045	. 2 class VtxDeg
2		vg	. . 3 setvar 𝑔
3		cvv 2799	. . 3 class V
4		vv	. . . 4 setvar 𝑣
5	2	cv 1394	. . . . 5 class 𝑔
6		cvtx 15828	. . . . 5 class Vtx
7	5, 6	cfv 5318	. . . 4 class (Vtx‘𝑔)
8		ve	. . . . 5 setvar 𝑒
9		ciedg 15829	. . . . . 6 class iEdg
10	5, 9	cfv 5318	. . . . 5 class (iEdg‘𝑔)
11		vu	. . . . . 6 setvar 𝑢
12	4	cv 1394	. . . . . 6 class 𝑣
13	11	cv 1394	. . . . . . . . . 10 class 𝑢
14		vx	. . . . . . . . . . . 12 setvar 𝑥
15	14	cv 1394	. . . . . . . . . . 11 class 𝑥
16	8	cv 1394	. . . . . . . . . . 11 class 𝑒
17	15, 16	cfv 5318	. . . . . . . . . 10 class (𝑒‘𝑥)
18	13, 17	wcel 2200	. . . . . . . . 9 wff 𝑢 ∈ (𝑒‘𝑥)
19	16	cdm 4719	. . . . . . . . 9 class dom 𝑒
20	18, 14, 19	crab 2512	. . . . . . . 8 class {𝑥 ∈ dom 𝑒 ∣ 𝑢 ∈ (𝑒‘𝑥)}
21		chash 11009	. . . . . . . 8 class ♯
22	20, 21	cfv 5318	. . . . . . 7 class (♯‘{𝑥 ∈ dom 𝑒 ∣ 𝑢 ∈ (𝑒‘𝑥)})
23	13	csn 3666	. . . . . . . . . 10 class {𝑢}
24	17, 23	wceq 1395	. . . . . . . . 9 wff (𝑒‘𝑥) = {𝑢}
25	24, 14, 19	crab 2512	. . . . . . . 8 class {𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) = {𝑢}}
26	25, 21	cfv 5318	. . . . . . 7 class (♯‘{𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) = {𝑢}})
27		cxad 9978	. . . . . . 7 class +𝑒
28	22, 26, 27	co 6007	. . . . . 6 class ((♯‘{𝑥 ∈ dom 𝑒 ∣ 𝑢 ∈ (𝑒‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) = {𝑢}}))
29	11, 12, 28	cmpt 4145	. . . . 5 class (𝑢 ∈ 𝑣 ↦ ((♯‘{𝑥 ∈ dom 𝑒 ∣ 𝑢 ∈ (𝑒‘𝑥)}) +𝑒 (♯‘{𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) = {𝑢}})))
30	8, 10, 29	csb 3124	. . . 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 3124	. . 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 4145	. 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 1395	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  16047