Definition df-vtxdg 16408

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 7179), 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 e. _V |-> [_ (Vtx ` g ) / v ]_ [_ (iEdg ` g ) / e ]_ ( u e. v |-> ( (♯ ` { x e. dom e | u e. ( e ` x ) } ) +e (♯ ` { x e. 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 16407	. 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 16133	. . . . 5 class Vtx
7	5, 6	cfv 5357	. . . 4 class (Vtx ` g )
8		ve	. . . . 5 setvar e
9		ciedg 16134	. . . . . 6 class iEdg
10	5, 9	cfv 5357	. . . . 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 5357	. . . . . . . . . 10 class ( e ` x )
18	13, 17	wcel 2205	. . . . . . . . 9 wff u e. ( e ` x )
19	16	cdm 4754	. . . . . . . . 9 class dom e
20	18, 14, 19	crab 2526	. . . . . . . 8 class { x e. dom e | u e. ( e ` x ) }
21		chash 11163	. . . . . . . 8 class ♯
22	20, 21	cfv 5357	. . . . . . 7 class (♯ ` { x e. dom e | u e. ( e ` x ) } )
23	13	csn 3694	. . . . . . . . . 10 class { u }
24	17, 23	wceq 1398	. . . . . . . . 9 wff ( e ` x ) = { u }
25	24, 14, 19	crab 2526	. . . . . . . 8 class { x e. dom e | ( e ` x ) = { u } }
26	25, 21	cfv 5357	. . . . . . 7 class (♯ ` { x e. dom e | ( e ` x ) = { u } } )
27		cxad 10122	. . . . . . 7 class +e
28	22, 26, 27	co 6058	. . . . . 6 class ( (♯ ` { x e. dom e | u e. ( e ` x ) } ) +e (♯ ` { x e. dom e | ( e ` x ) = { u } } ) )
29	11, 12, 28	cmpt 4176	. . . . 5 class ( u e. v |-> ( (♯ ` { x e. dom e | u e. ( e ` x ) } ) +e (♯ ` { x e. dom e | ( e ` x ) = { u } } ) ) )
30	8, 10, 29	csb 3141	. . . 4 class [_ (iEdg ` g ) / e ]_ ( u e. v |-> ( (♯ ` { x e. dom e | u e. ( e ` x ) } ) +e (♯ ` { x e. dom e | ( e ` x ) = { u } } ) ) )
31	4, 7, 30	csb 3141	. . 3 class [_ (Vtx ` g ) / v ]_ [_ (iEdg ` g ) / e ]_ ( u e. v |-> ( (♯ ` { x e. dom e | u e. ( e ` x ) } ) +e (♯ ` { x e. dom e | ( e ` x ) = { u } } ) ) )
32	2, 3, 31	cmpt 4176	. 2 class ( g e. _V |-> [_ (Vtx ` g ) / v ]_ [_ (iEdg ` g ) / e ]_ ( u e. v |-> ( (♯ ` { x e. dom e | u e. ( e ` x ) } ) +e (♯ ` { x e. dom e | ( e ` x ) = { u } } ) ) ) )
33	1, 32	wceq 1398	1 wff VtxDeg = ( g e. _V |-> [_ (Vtx ` g ) / v ]_ [_ (iEdg ` g ) / e ]_ ( u e. v |-> ( (♯ ` { x e. dom e | u e. ( e ` x ) } ) +e (♯ ` { x e. dom e | ( e ` x ) = { u } } ) ) ) )

This definition is referenced by:  vtxdgfval  16409