Definition df-vtxdg 16282

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 7166), 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 16281	. 2 class VtxDeg
2		vg	. . 3 setvar g
3		cvv 2813	. . 3 class _V
4		vv	. . . 4 setvar v
5	2	cv 1397	. . . . 5 class g
6		cvtx 16007	. . . . 5 class Vtx
7	5, 6	cfv 5352	. . . 4 class (Vtx ` g )
8		ve	. . . . 5 setvar e
9		ciedg 16008	. . . . . 6 class iEdg
10	5, 9	cfv 5352	. . . . 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 5352	. . . . . . . . . 10 class ( e ` x )
18	13, 17	wcel 2203	. . . . . . . . 9 wff u e. ( e ` x )
19	16	cdm 4749	. . . . . . . . 9 class dom e
20	18, 14, 19	crab 2524	. . . . . . . 8 class { x e. dom e | u e. ( e ` x ) }
21		chash 11138	. . . . . . . 8 class ♯
22	20, 21	cfv 5352	. . . . . . 7 class (♯ ` { x e. dom e | u e. ( e ` x ) } )
23	13	csn 3689	. . . . . . . . . 10 class { u }
24	17, 23	wceq 1398	. . . . . . . . 9 wff ( e ` x ) = { u }
25	24, 14, 19	crab 2524	. . . . . . . 8 class { x e. dom e | ( e ` x ) = { u } }
26	25, 21	cfv 5352	. . . . . . 7 class (♯ ` { x e. dom e | ( e ` x ) = { u } } )
27		cxad 10103	. . . . . . 7 class +e
28	22, 26, 27	co 6050	. . . . . 6 class ( (♯ ` { x e. dom e | u e. ( e ` x ) } ) +e (♯ ` { x e. dom e | ( e ` x ) = { u } } ) )
29	11, 12, 28	cmpt 4171	. . . . 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 3138	. . . 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 3138	. . 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 4171	. 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  16283