Definition df-vtxdg 16442

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 7203), 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 16441	. 2 class VtxDeg
2		vg	. . 3 setvar g
3		cvv 2821	. . 3 class _V
4		vv	. . . 4 setvar v
5	2	cv 1401	. . . . 5 class g
6		cvtx 16167	. . . . 5 class Vtx
7	5, 6	cfv 5372	. . . 4 class (Vtx ` g )
8		ve	. . . . 5 setvar e
9		ciedg 16168	. . . . . 6 class iEdg
10	5, 9	cfv 5372	. . . . 5 class (iEdg ` g )
11		vu	. . . . . 6 setvar u
12	4	cv 1401	. . . . . 6 class v
13	11	cv 1401	. . . . . . . . . 10 class u
14		vx	. . . . . . . . . . . 12 setvar x
15	14	cv 1401	. . . . . . . . . . 11 class x
16	8	cv 1401	. . . . . . . . . . 11 class e
17	15, 16	cfv 5372	. . . . . . . . . 10 class ( e ` x )
18	13, 17	wcel 2209	. . . . . . . . 9 wff u e. ( e ` x )
19	16	cdm 4769	. . . . . . . . 9 class dom e
20	18, 14, 19	crab 2532	. . . . . . . 8 class { x e. dom e | u e. ( e ` x ) }
21		chash 11192	. . . . . . . 8 class ♯
22	20, 21	cfv 5372	. . . . . . 7 class (♯ ` { x e. dom e | u e. ( e ` x ) } )
23	13	csn 3705	. . . . . . . . . 10 class { u }
24	17, 23	wceq 1402	. . . . . . . . 9 wff ( e ` x ) = { u }
25	24, 14, 19	crab 2532	. . . . . . . 8 class { x e. dom e | ( e ` x ) = { u } }
26	25, 21	cfv 5372	. . . . . . 7 class (♯ ` { x e. dom e | ( e ` x ) = { u } } )
27		cxad 10151	. . . . . . 7 class +e
28	22, 26, 27	co 6075	. . . . . 6 class ( (♯ ` { x e. dom e | u e. ( e ` x ) } ) +e (♯ ` { x e. dom e | ( e ` x ) = { u } } ) )
29	11, 12, 28	cmpt 4187	. . . . 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 3147	. . . 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 3147	. . 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 4187	. 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 1402	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  16443