Theorem fusgr2wsp2nb 27198

Description: The set of paths of length 2 with a given vertex in the middle for a finite simple graph is the union of all paths of length 2 from one neighbor to another neighbor of this vertex via this vertex. (Contributed by Alexander van der Vekens, 9-Mar-2018.) (Revised by AV, 17-May-2021.) (Proof shortened by AV, 8-Jan-2022.)

Hypotheses

Ref	Expression
frgrhash2wsp.v	|- V = (Vtx ` G )
fusgreg2wsp.m	|- M = ( a e. V |-> { w e. ( 2 WSPathsN G ) | ( w ` 1 ) = a } )

Assertion

Ref	Expression
fusgr2wsp2nb	|- ( ( G e. FinUSGraph /\ N e. V ) -> ( M ` N ) = U_ x e. ( G NeighbVtx N ) U_ y e. ( ( G NeighbVtx N ) \ { x } ) { <" x N y "> } )

Distinct variable groups:   G, a   V, a   w, G   N, a, w   x, G, y   x, N, y   x, V, y

Allowed substitution hints:   M( x, y, w, a)   V( w)

Proof of Theorem fusgr2wsp2nb

Dummy variables m z p are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		frgrhash2wsp.v	. . . . . 6 |- V = (Vtx ` G )
2		fusgreg2wsp.m	. . . . . 6 |- M = ( a e. V |-> { w e. ( 2 WSPathsN G ) | ( w ` 1 ) = a } )
3	1, 2	fusgreg2wsplem 27197	. . . . 5 |- ( N e. V -> ( z e. ( M ` N ) <-> ( z e. ( 2 WSPathsN G ) /\ ( z ` 1 ) = N ) ) )
4	3	adantl 482	. . . 4 |- ( ( G e. FinUSGraph /\ N e. V ) -> ( z e. ( M ` N ) <-> ( z e. ( 2 WSPathsN G ) /\ ( z ` 1 ) = N ) ) )
5		2nn0 11309	. . . . . . . . 9 |- 2 e. NN0
6	1	wspthsnwspthsnon 26811	. . . . . . . . 9 |- ( ( 2 e. NN0 /\ G e. FinUSGraph ) -> ( z e. ( 2 WSPathsN G ) <-> E. x e. V E. y e. V z e. ( x ( 2 WSPathsNOn G ) y ) ) )
7	5, 6	mpan 706	. . . . . . . 8 |- ( G e. FinUSGraph -> ( z e. ( 2 WSPathsN G ) <-> E. x e. V E. y e. V z e. ( x ( 2 WSPathsNOn G ) y ) ) )
8	7	adantr 481	. . . . . . 7 |- ( ( G e. FinUSGraph /\ N e. V ) -> ( z e. ( 2 WSPathsN G ) <-> E. x e. V E. y e. V z e. ( x ( 2 WSPathsNOn G ) y ) ) )
9		fusgrusgr 26214	. . . . . . . . . 10 |- ( G e. FinUSGraph -> G e. USGraph )
10	9	adantr 481	. . . . . . . . 9 |- ( ( G e. FinUSGraph /\ N e. V ) -> G e. USGraph )
11		eqid 2622	. . . . . . . . . 10 |- (Edg ` G ) = (Edg ` G )
12	1, 11	usgr2wspthon 26858	. . . . . . . . 9 |- ( ( G e. USGraph /\ ( x e. V /\ y e. V ) ) -> ( z e. ( x ( 2 WSPathsNOn G ) y ) <-> E. m e. V ( ( z = <" x m y "> /\ x =/= y ) /\ ( { x , m } e. (Edg ` G ) /\ { m , y } e. (Edg ` G ) ) ) ) )
13	10, 12	sylan 488	. . . . . . . 8 |- ( ( ( G e. FinUSGraph /\ N e. V ) /\ ( x e. V /\ y e. V ) ) -> ( z e. ( x ( 2 WSPathsNOn G ) y ) <-> E. m e. V ( ( z = <" x m y "> /\ x =/= y ) /\ ( { x , m } e. (Edg ` G ) /\ { m , y } e. (Edg ` G ) ) ) ) )
14	13	2rexbidva 3056	. . . . . . 7 |- ( ( G e. FinUSGraph /\ N e. V ) -> ( E. x e. V E. y e. V z e. ( x ( 2 WSPathsNOn G ) y ) <-> E. x e. V E. y e. V E. m e. V ( ( z = <" x m y "> /\ x =/= y ) /\ ( { x , m } e. (Edg ` G ) /\ { m , y } e. (Edg ` G ) ) ) ) )
15	8, 14	bitrd 268	. . . . . 6 |- ( ( G e. FinUSGraph /\ N e. V ) -> ( z e. ( 2 WSPathsN G ) <-> E. x e. V E. y e. V E. m e. V ( ( z = <" x m y "> /\ x =/= y ) /\ ( { x , m } e. (Edg ` G ) /\ { m , y } e. (Edg ` G ) ) ) ) )
16	15	anbi1d 741	. . . . 5 |- ( ( G e. FinUSGraph /\ N e. V ) -> ( ( z e. ( 2 WSPathsN G ) /\ ( z ` 1 ) = N ) <-> ( E. x e. V E. y e. V E. m e. V ( ( z = <" x m y "> /\ x =/= y ) /\ ( { x , m } e. (Edg ` G ) /\ { m , y } e. (Edg ` G ) ) ) /\ ( z ` 1 ) = N ) ) )
17		19.41vv 1915	. . . . . . 7 |- ( E. x E. y ( ( ( x e. V /\ y e. V ) /\ E. m e. V ( ( z = <" x m y "> /\ x =/= y ) /\ ( { x , m } e. (Edg ` G ) /\ { m , y } e. (Edg ` G ) ) ) ) /\ ( z ` 1 ) = N ) <-> ( E. x E. y ( ( x e. V /\ y e. V ) /\ E. m e. V ( ( z = <" x m y "> /\ x =/= y ) /\ ( { x , m } e. (Edg ` G ) /\ { m , y } e. (Edg ` G ) ) ) ) /\ ( z ` 1 ) = N ) )
18		velsn 4193	. . . . . . . . . . . 12 |- ( z e. { <" x N y "> } <-> z = <" x N y "> )
19	18	bicomi 214	. . . . . . . . . . 11 |- ( z = <" x N y "> <-> z e. { <" x N y "> } )
20	19	anbi2i 730	. . . . . . . . . 10 |- ( ( ( { x , N } e. (Edg ` G ) /\ ( { y , N } e. (Edg ` G ) /\ -. y = x ) ) /\ z = <" x N y "> ) <-> ( ( { x , N } e. (Edg ` G ) /\ ( { y , N } e. (Edg ` G ) /\ -. y = x ) ) /\ z e. { <" x N y "> } ) )
21	20	a1i 11	. . . . . . . . 9 |- ( ( G e. FinUSGraph /\ N e. V ) -> ( ( ( { x , N } e. (Edg ` G ) /\ ( { y , N } e. (Edg ` G ) /\ -. y = x ) ) /\ z = <" x N y "> ) <-> ( ( { x , N } e. (Edg ` G ) /\ ( { y , N } e. (Edg ` G ) /\ -. y = x ) ) /\ z e. { <" x N y "> } ) ) )
22		simplr 792	. . . . . . . . . . . 12 |- ( ( ( G e. FinUSGraph /\ N e. V ) /\ ( ( x e. V /\ y e. V ) /\ ( z ` 1 ) = N ) ) -> N e. V )
23		anass 681	. . . . . . . . . . . . . . 15 |- ( ( ( z = <" x m y "> /\ x =/= y ) /\ ( { x , m } e. (Edg ` G ) /\ { m , y } e. (Edg ` G ) ) ) <-> ( z = <" x m y "> /\ ( x =/= y /\ ( { x , m } e. (Edg ` G ) /\ { m , y } e. (Edg ` G ) ) ) ) )
24		ancom 466	. . . . . . . . . . . . . . 15 |- ( ( z = <" x m y "> /\ ( x =/= y /\ ( { x , m } e. (Edg ` G ) /\ { m , y } e. (Edg ` G ) ) ) ) <-> ( ( x =/= y /\ ( { x , m } e. (Edg ` G ) /\ { m , y } e. (Edg ` G ) ) ) /\ z = <" x m y "> ) )
25		an12 838	. . . . . . . . . . . . . . . . 17 |- ( ( x =/= y /\ ( { x , m } e. (Edg ` G ) /\ { m , y } e. (Edg ` G ) ) ) <-> ( { x , m } e. (Edg ` G ) /\ ( x =/= y /\ { m , y } e. (Edg ` G ) ) ) )
26		nesym 2850	. . . . . . . . . . . . . . . . . . 19 |- ( x =/= y <-> -. y = x )
27		prcom 4267	. . . . . . . . . . . . . . . . . . . 20 |- { m , y } = { y , m }
28	27	eleq1i 2692	. . . . . . . . . . . . . . . . . . 19 |- ( { m , y } e. (Edg ` G ) <-> { y , m } e. (Edg ` G ) )
29	26, 28	anbi12ci 734	. . . . . . . . . . . . . . . . . 18 |- ( ( x =/= y /\ { m , y } e. (Edg ` G ) ) <-> ( { y , m } e. (Edg ` G ) /\ -. y = x ) )
30	29	anbi2i 730	. . . . . . . . . . . . . . . . 17 |- ( ( { x , m } e. (Edg ` G ) /\ ( x =/= y /\ { m , y } e. (Edg ` G ) ) ) <-> ( { x , m } e. (Edg ` G ) /\ ( { y , m } e. (Edg ` G ) /\ -. y = x ) ) )
31	25, 30	bitri 264	. . . . . . . . . . . . . . . 16 |- ( ( x =/= y /\ ( { x , m } e. (Edg ` G ) /\ { m , y } e. (Edg ` G ) ) ) <-> ( { x , m } e. (Edg ` G ) /\ ( { y , m } e. (Edg ` G ) /\ -. y = x ) ) )
32	31	anbi1i 731	. . . . . . . . . . . . . . 15 |- ( ( ( x =/= y /\ ( { x , m } e. (Edg ` G ) /\ { m , y } e. (Edg ` G ) ) ) /\ z = <" x m y "> ) <-> ( ( { x , m } e. (Edg ` G ) /\ ( { y , m } e. (Edg ` G ) /\ -. y = x ) ) /\ z = <" x m y "> ) )
33	23, 24, 32	3bitri 286	. . . . . . . . . . . . . 14 |- ( ( ( z = <" x m y "> /\ x =/= y ) /\ ( { x , m } e. (Edg ` G ) /\ { m , y } e. (Edg ` G ) ) ) <-> ( ( { x , m } e. (Edg ` G ) /\ ( { y , m } e. (Edg ` G ) /\ -. y = x ) ) /\ z = <" x m y "> ) )
34		preq2 4269	. . . . . . . . . . . . . . . . 17 |- ( m = N -> { x , m } = { x , N } )
35	34	eleq1d 2686	. . . . . . . . . . . . . . . 16 |- ( m = N -> ( { x , m } e. (Edg ` G ) <-> { x , N } e. (Edg ` G ) ) )
36		preq2 4269	. . . . . . . . . . . . . . . . . 18 |- ( m = N -> { y , m } = { y , N } )
37	36	eleq1d 2686	. . . . . . . . . . . . . . . . 17 |- ( m = N -> ( { y , m } e. (Edg ` G ) <-> { y , N } e. (Edg ` G ) ) )
38	37	anbi1d 741	. . . . . . . . . . . . . . . 16 |- ( m = N -> ( ( { y , m } e. (Edg ` G ) /\ -. y = x ) <-> ( { y , N } e. (Edg ` G ) /\ -. y = x ) ) )
39	35, 38	anbi12d 747	. . . . . . . . . . . . . . 15 |- ( m = N -> ( ( { x , m } e. (Edg ` G ) /\ ( { y , m } e. (Edg ` G ) /\ -. y = x ) ) <-> ( { x , N } e. (Edg ` G ) /\ ( { y , N } e. (Edg ` G ) /\ -. y = x ) ) ) )
40		s3eq2 13615	. . . . . . . . . . . . . . . 16 |- ( m = N -> <" x m y "> = <" x N y "> )
41	40	eqeq2d 2632	. . . . . . . . . . . . . . 15 |- ( m = N -> ( z = <" x m y "> <-> z = <" x N y "> ) )
42	39, 41	anbi12d 747	. . . . . . . . . . . . . 14 |- ( m = N -> ( ( ( { x , m } e. (Edg ` G ) /\ ( { y , m } e. (Edg ` G ) /\ -. y = x ) ) /\ z = <" x m y "> ) <-> ( ( { x , N } e. (Edg ` G ) /\ ( { y , N } e. (Edg ` G ) /\ -. y = x ) ) /\ z = <" x N y "> ) ) )
43	33, 42	syl5bb 272	. . . . . . . . . . . . 13 |- ( m = N -> ( ( ( z = <" x m y "> /\ x =/= y ) /\ ( { x , m } e. (Edg ` G ) /\ { m , y } e. (Edg ` G ) ) ) <-> ( ( { x , N } e. (Edg ` G ) /\ ( { y , N } e. (Edg ` G ) /\ -. y = x ) ) /\ z = <" x N y "> ) ) )
44	43	adantl 482	. . . . . . . . . . . 12 |- ( ( ( ( G e. FinUSGraph /\ N e. V ) /\ ( ( x e. V /\ y e. V ) /\ ( z ` 1 ) = N ) ) /\ m = N ) -> ( ( ( z = <" x m y "> /\ x =/= y ) /\ ( { x , m } e. (Edg ` G ) /\ { m , y } e. (Edg ` G ) ) ) <-> ( ( { x , N } e. (Edg ` G ) /\ ( { y , N } e. (Edg ` G ) /\ -. y = x ) ) /\ z = <" x N y "> ) ) )
45		fveq1 6190	. . . . . . . . . . . . . . . . . . . 20 |- ( z = <" x m y "> -> ( z ` 1 ) = ( <" x m y "> ` 1 ) )
46		vex 3203	. . . . . . . . . . . . . . . . . . . . 21 |- m e. _V
47		s3fv1 13637	. . . . . . . . . . . . . . . . . . . . 21 |- ( m e. _V -> ( <" x m y "> ` 1 ) = m )
48	46, 47	ax-mp 5	. . . . . . . . . . . . . . . . . . . 20 |- ( <" x m y "> ` 1 ) = m
49	45, 48	syl6eq 2672	. . . . . . . . . . . . . . . . . . 19 |- ( z = <" x m y "> -> ( z ` 1 ) = m )
50	49	eqeq1d 2624	. . . . . . . . . . . . . . . . . 18 |- ( z = <" x m y "> -> ( ( z ` 1 ) = N <-> m = N ) )
51	50	biimpd 219	. . . . . . . . . . . . . . . . 17 |- ( z = <" x m y "> -> ( ( z ` 1 ) = N -> m = N ) )
52	51	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( z = <" x m y "> /\ x =/= y ) -> ( ( z ` 1 ) = N -> m = N ) )
53	52	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ( z = <" x m y "> /\ x =/= y ) /\ ( { x , m } e. (Edg ` G ) /\ { m , y } e. (Edg ` G ) ) ) -> ( ( z ` 1 ) = N -> m = N ) )
54	53	com12 32	. . . . . . . . . . . . . 14 |- ( ( z ` 1 ) = N -> ( ( ( z = <" x m y "> /\ x =/= y ) /\ ( { x , m } e. (Edg ` G ) /\ { m , y } e. (Edg ` G ) ) ) -> m = N ) )
55	54	ad2antll 765	. . . . . . . . . . . . 13 |- ( ( ( G e. FinUSGraph /\ N e. V ) /\ ( ( x e. V /\ y e. V ) /\ ( z ` 1 ) = N ) ) -> ( ( ( z = <" x m y "> /\ x =/= y ) /\ ( { x , m } e. (Edg ` G ) /\ { m , y } e. (Edg ` G ) ) ) -> m = N ) )
56	55	imp 445	. . . . . . . . . . . 12 |- ( ( ( ( G e. FinUSGraph /\ N e. V ) /\ ( ( x e. V /\ y e. V ) /\ ( z ` 1 ) = N ) ) /\ ( ( z = <" x m y "> /\ x =/= y ) /\ ( { x , m } e. (Edg ` G ) /\ { m , y } e. (Edg ` G ) ) ) ) -> m = N )
57	22, 44, 56	rspcebdv 3314	. . . . . . . . . . 11 |- ( ( ( G e. FinUSGraph /\ N e. V ) /\ ( ( x e. V /\ y e. V ) /\ ( z ` 1 ) = N ) ) -> ( E. m e. V ( ( z = <" x m y "> /\ x =/= y ) /\ ( { x , m } e. (Edg ` G ) /\ { m , y } e. (Edg ` G ) ) ) <-> ( ( { x , N } e. (Edg ` G ) /\ ( { y , N } e. (Edg ` G ) /\ -. y = x ) ) /\ z = <" x N y "> ) ) )
58	57	pm5.32da 673	. . . . . . . . . 10 |- ( ( G e. FinUSGraph /\ N e. V ) -> ( ( ( ( x e. V /\ y e. V ) /\ ( z ` 1 ) = N ) /\ E. m e. V ( ( z = <" x m y "> /\ x =/= y ) /\ ( { x , m } e. (Edg ` G ) /\ { m , y } e. (Edg ` G ) ) ) ) <-> ( ( ( x e. V /\ y e. V ) /\ ( z ` 1 ) = N ) /\ ( ( { x , N } e. (Edg ` G ) /\ ( { y , N } e. (Edg ` G ) /\ -. y = x ) ) /\ z = <" x N y "> ) ) ) )
59		an32 839	. . . . . . . . . . 11 |- ( ( ( ( x e. V /\ y e. V ) /\ E. m e. V ( ( z = <" x m y "> /\ x =/= y ) /\ ( { x , m } e. (Edg ` G ) /\ { m , y } e. (Edg ` G ) ) ) ) /\ ( z ` 1 ) = N ) <-> ( ( ( x e. V /\ y e. V ) /\ ( z ` 1 ) = N ) /\ E. m e. V ( ( z = <" x m y "> /\ x =/= y ) /\ ( { x , m } e. (Edg ` G ) /\ { m , y } e. (Edg ` G ) ) ) ) )
60	59	a1i 11	. . . . . . . . . 10 |- ( ( G e. FinUSGraph /\ N e. V ) -> ( ( ( ( x e. V /\ y e. V ) /\ E. m e. V ( ( z = <" x m y "> /\ x =/= y ) /\ ( { x , m } e. (Edg ` G ) /\ { m , y } e. (Edg ` G ) ) ) ) /\ ( z ` 1 ) = N ) <-> ( ( ( x e. V /\ y e. V ) /\ ( z ` 1 ) = N ) /\ E. m e. V ( ( z = <" x m y "> /\ x =/= y ) /\ ( { x , m } e. (Edg ` G ) /\ { m , y } e. (Edg ` G ) ) ) ) ) )
61		usgrumgr 26074	. . . . . . . . . . . . . . . . . 18 |- ( G e. USGraph -> G e. UMGraph )
62	1, 11	umgrpredgv 26035	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( G e. UMGraph /\ { x , N } e. (Edg ` G ) ) -> ( x e. V /\ N e. V ) )
63	62	simpld 475	. . . . . . . . . . . . . . . . . . . 20 |- ( ( G e. UMGraph /\ { x , N } e. (Edg ` G ) ) -> x e. V )
64	63	ex 450	. . . . . . . . . . . . . . . . . . 19 |- ( G e. UMGraph -> ( { x , N } e. (Edg ` G ) -> x e. V ) )
65	1, 11	umgrpredgv 26035	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( G e. UMGraph /\ { y , N } e. (Edg ` G ) ) -> ( y e. V /\ N e. V ) )
66	65	simpld 475	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( G e. UMGraph /\ { y , N } e. (Edg ` G ) ) -> y e. V )
67	66	expcom 451	. . . . . . . . . . . . . . . . . . . . 21 |- ( { y , N } e. (Edg ` G ) -> ( G e. UMGraph -> y e. V ) )
68	67	adantr 481	. . . . . . . . . . . . . . . . . . . 20 |- ( ( { y , N } e. (Edg ` G ) /\ -. y = x ) -> ( G e. UMGraph -> y e. V ) )
69	68	com12 32	. . . . . . . . . . . . . . . . . . 19 |- ( G e. UMGraph -> ( ( { y , N } e. (Edg ` G ) /\ -. y = x ) -> y e. V ) )
70	64, 69	anim12d 586	. . . . . . . . . . . . . . . . . 18 |- ( G e. UMGraph -> ( ( { x , N } e. (Edg ` G ) /\ ( { y , N } e. (Edg ` G ) /\ -. y = x ) ) -> ( x e. V /\ y e. V ) ) )
71	9, 61, 70	3syl 18	. . . . . . . . . . . . . . . . 17 |- ( G e. FinUSGraph -> ( ( { x , N } e. (Edg ` G ) /\ ( { y , N } e. (Edg ` G ) /\ -. y = x ) ) -> ( x e. V /\ y e. V ) ) )
72	71	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( G e. FinUSGraph /\ N e. V ) -> ( ( { x , N } e. (Edg ` G ) /\ ( { y , N } e. (Edg ` G ) /\ -. y = x ) ) -> ( x e. V /\ y e. V ) ) )
73	72	com12 32	. . . . . . . . . . . . . . 15 |- ( ( { x , N } e. (Edg ` G ) /\ ( { y , N } e. (Edg ` G ) /\ -. y = x ) ) -> ( ( G e. FinUSGraph /\ N e. V ) -> ( x e. V /\ y e. V ) ) )
74	73	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ( { x , N } e. (Edg ` G ) /\ ( { y , N } e. (Edg ` G ) /\ -. y = x ) ) /\ z = <" x N y "> ) -> ( ( G e. FinUSGraph /\ N e. V ) -> ( x e. V /\ y e. V ) ) )
75	74	impcom 446	. . . . . . . . . . . . 13 |- ( ( ( G e. FinUSGraph /\ N e. V ) /\ ( ( { x , N } e. (Edg ` G ) /\ ( { y , N } e. (Edg ` G ) /\ -. y = x ) ) /\ z = <" x N y "> ) ) -> ( x e. V /\ y e. V ) )
76		fveq1 6190	. . . . . . . . . . . . . . 15 |- ( z = <" x N y "> -> ( z ` 1 ) = ( <" x N y "> ` 1 ) )
77	76	adantl 482	. . . . . . . . . . . . . 14 |- ( ( ( { x , N } e. (Edg ` G ) /\ ( { y , N } e. (Edg ` G ) /\ -. y = x ) ) /\ z = <" x N y "> ) -> ( z ` 1 ) = ( <" x N y "> ` 1 ) )
78		s3fv1 13637	. . . . . . . . . . . . . . 15 |- ( N e. V -> ( <" x N y "> ` 1 ) = N )
79	78	adantl 482	. . . . . . . . . . . . . 14 |- ( ( G e. FinUSGraph /\ N e. V ) -> ( <" x N y "> ` 1 ) = N )
80	77, 79	sylan9eqr 2678	. . . . . . . . . . . . 13 |- ( ( ( G e. FinUSGraph /\ N e. V ) /\ ( ( { x , N } e. (Edg ` G ) /\ ( { y , N } e. (Edg ` G ) /\ -. y = x ) ) /\ z = <" x N y "> ) ) -> ( z ` 1 ) = N )
81	75, 80	jca 554	. . . . . . . . . . . 12 |- ( ( ( G e. FinUSGraph /\ N e. V ) /\ ( ( { x , N } e. (Edg ` G ) /\ ( { y , N } e. (Edg ` G ) /\ -. y = x ) ) /\ z = <" x N y "> ) ) -> ( ( x e. V /\ y e. V ) /\ ( z ` 1 ) = N ) )
82	81	ex 450	. . . . . . . . . . 11 |- ( ( G e. FinUSGraph /\ N e. V ) -> ( ( ( { x , N } e. (Edg ` G ) /\ ( { y , N } e. (Edg ` G ) /\ -. y = x ) ) /\ z = <" x N y "> ) -> ( ( x e. V /\ y e. V ) /\ ( z ` 1 ) = N ) ) )
83	82	pm4.71rd 667	. . . . . . . . . 10 |- ( ( G e. FinUSGraph /\ N e. V ) -> ( ( ( { x , N } e. (Edg ` G ) /\ ( { y , N } e. (Edg ` G ) /\ -. y = x ) ) /\ z = <" x N y "> ) <-> ( ( ( x e. V /\ y e. V ) /\ ( z ` 1 ) = N ) /\ ( ( { x , N } e. (Edg ` G ) /\ ( { y , N } e. (Edg ` G ) /\ -. y = x ) ) /\ z = <" x N y "> ) ) ) )
84	58, 60, 83	3bitr4d 300	. . . . . . . . 9 |- ( ( G e. FinUSGraph /\ N e. V ) -> ( ( ( ( x e. V /\ y e. V ) /\ E. m e. V ( ( z = <" x m y "> /\ x =/= y ) /\ ( { x , m } e. (Edg ` G ) /\ { m , y } e. (Edg ` G ) ) ) ) /\ ( z ` 1 ) = N ) <-> ( ( { x , N } e. (Edg ` G ) /\ ( { y , N } e. (Edg ` G ) /\ -. y = x ) ) /\ z = <" x N y "> ) ) )
85	11	nbusgreledg 26249	. . . . . . . . . . . . 13 |- ( G e. USGraph -> ( x e. ( G NeighbVtx N ) <-> { x , N } e. (Edg ` G ) ) )
86	9, 85	syl 17	. . . . . . . . . . . 12 |- ( G e. FinUSGraph -> ( x e. ( G NeighbVtx N ) <-> { x , N } e. (Edg ` G ) ) )
87	86	adantr 481	. . . . . . . . . . 11 |- ( ( G e. FinUSGraph /\ N e. V ) -> ( x e. ( G NeighbVtx N ) <-> { x , N } e. (Edg ` G ) ) )
88		eldif 3584	. . . . . . . . . . . 12 |- ( y e. ( ( G NeighbVtx N ) \ { x } ) <-> ( y e. ( G NeighbVtx N ) /\ -. y e. { x } ) )
89	11	nbusgreledg 26249	. . . . . . . . . . . . . . 15 |- ( G e. USGraph -> ( y e. ( G NeighbVtx N ) <-> { y , N } e. (Edg ` G ) ) )
90	9, 89	syl 17	. . . . . . . . . . . . . 14 |- ( G e. FinUSGraph -> ( y e. ( G NeighbVtx N ) <-> { y , N } e. (Edg ` G ) ) )
91	90	adantr 481	. . . . . . . . . . . . 13 |- ( ( G e. FinUSGraph /\ N e. V ) -> ( y e. ( G NeighbVtx N ) <-> { y , N } e. (Edg ` G ) ) )
92		velsn 4193	. . . . . . . . . . . . . . 15 |- ( y e. { x } <-> y = x )
93	92	a1i 11	. . . . . . . . . . . . . 14 |- ( ( G e. FinUSGraph /\ N e. V ) -> ( y e. { x } <-> y = x ) )
94	93	notbid 308	. . . . . . . . . . . . 13 |- ( ( G e. FinUSGraph /\ N e. V ) -> ( -. y e. { x } <-> -. y = x ) )
95	91, 94	anbi12d 747	. . . . . . . . . . . 12 |- ( ( G e. FinUSGraph /\ N e. V ) -> ( ( y e. ( G NeighbVtx N ) /\ -. y e. { x } ) <-> ( { y , N } e. (Edg ` G ) /\ -. y = x ) ) )
96	88, 95	syl5bb 272	. . . . . . . . . . 11 |- ( ( G e. FinUSGraph /\ N e. V ) -> ( y e. ( ( G NeighbVtx N ) \ { x } ) <-> ( { y , N } e. (Edg ` G ) /\ -. y = x ) ) )
97	87, 96	anbi12d 747	. . . . . . . . . 10 |- ( ( G e. FinUSGraph /\ N e. V ) -> ( ( x e. ( G NeighbVtx N ) /\ y e. ( ( G NeighbVtx N ) \ { x } ) ) <-> ( { x , N } e. (Edg ` G ) /\ ( { y , N } e. (Edg ` G ) /\ -. y = x ) ) ) )
98	97	anbi1d 741	. . . . . . . . 9 |- ( ( G e. FinUSGraph /\ N e. V ) -> ( ( ( x e. ( G NeighbVtx N ) /\ y e. ( ( G NeighbVtx N ) \ { x } ) ) /\ z e. { <" x N y "> } ) <-> ( ( { x , N } e. (Edg ` G ) /\ ( { y , N } e. (Edg ` G ) /\ -. y = x ) ) /\ z e. { <" x N y "> } ) ) )
99	21, 84, 98	3bitr4d 300	. . . . . . . 8 |- ( ( G e. FinUSGraph /\ N e. V ) -> ( ( ( ( x e. V /\ y e. V ) /\ E. m e. V ( ( z = <" x m y "> /\ x =/= y ) /\ ( { x , m } e. (Edg ` G ) /\ { m , y } e. (Edg ` G ) ) ) ) /\ ( z ` 1 ) = N ) <-> ( ( x e. ( G NeighbVtx N ) /\ y e. ( ( G NeighbVtx N ) \ { x } ) ) /\ z e. { <" x N y "> } ) ) )
100	99	2exbidv 1852	. . . . . . 7 |- ( ( G e. FinUSGraph /\ N e. V ) -> ( E. x E. y ( ( ( x e. V /\ y e. V ) /\ E. m e. V ( ( z = <" x m y "> /\ x =/= y ) /\ ( { x , m } e. (Edg ` G ) /\ { m , y } e. (Edg ` G ) ) ) ) /\ ( z ` 1 ) = N ) <-> E. x E. y ( ( x e. ( G NeighbVtx N ) /\ y e. ( ( G NeighbVtx N ) \ { x } ) ) /\ z e. { <" x N y "> } ) ) )
101	17, 100	syl5bbr 274	. . . . . 6 |- ( ( G e. FinUSGraph /\ N e. V ) -> ( ( E. x E. y ( ( x e. V /\ y e. V ) /\ E. m e. V ( ( z = <" x m y "> /\ x =/= y ) /\ ( { x , m } e. (Edg ` G ) /\ { m , y } e. (Edg ` G ) ) ) ) /\ ( z ` 1 ) = N ) <-> E. x E. y ( ( x e. ( G NeighbVtx N ) /\ y e. ( ( G NeighbVtx N ) \ { x } ) ) /\ z e. { <" x N y "> } ) ) )
102		r2ex 3061	. . . . . . 7 |- ( E. x e. V E. y e. V E. m e. V ( ( z = <" x m y "> /\ x =/= y ) /\ ( { x , m } e. (Edg ` G ) /\ { m , y } e. (Edg ` G ) ) ) <-> E. x E. y ( ( x e. V /\ y e. V ) /\ E. m e. V ( ( z = <" x m y "> /\ x =/= y ) /\ ( { x , m } e. (Edg ` G ) /\ { m , y } e. (Edg ` G ) ) ) ) )
103	102	anbi1i 731	. . . . . 6 |- ( ( E. x e. V E. y e. V E. m e. V ( ( z = <" x m y "> /\ x =/= y ) /\ ( { x , m } e. (Edg ` G ) /\ { m , y } e. (Edg ` G ) ) ) /\ ( z ` 1 ) = N ) <-> ( E. x E. y ( ( x e. V /\ y e. V ) /\ E. m e. V ( ( z = <" x m y "> /\ x =/= y ) /\ ( { x , m } e. (Edg ` G ) /\ { m , y } e. (Edg ` G ) ) ) ) /\ ( z ` 1 ) = N ) )
104		r2ex 3061	. . . . . 6 |- ( E. x e. ( G NeighbVtx N ) E. y e. ( ( G NeighbVtx N ) \ { x } ) z e. { <" x N y "> } <-> E. x E. y ( ( x e. ( G NeighbVtx N ) /\ y e. ( ( G NeighbVtx N ) \ { x } ) ) /\ z e. { <" x N y "> } ) )
105	101, 103, 104	3bitr4g 303	. . . . 5 |- ( ( G e. FinUSGraph /\ N e. V ) -> ( ( E. x e. V E. y e. V E. m e. V ( ( z = <" x m y "> /\ x =/= y ) /\ ( { x , m } e. (Edg ` G ) /\ { m , y } e. (Edg ` G ) ) ) /\ ( z ` 1 ) = N ) <-> E. x e. ( G NeighbVtx N ) E. y e. ( ( G NeighbVtx N ) \ { x } ) z e. { <" x N y "> } ) )
106		vex 3203	. . . . . . . 8 |- z e. _V
107		eleq1w 2684	. . . . . . . . 9 |- ( p = z -> ( p e. { <" x N y "> } <-> z e. { <" x N y "> } ) )
108	107	2rexbidv 3057	. . . . . . . 8 |- ( p = z -> ( E. x e. ( G NeighbVtx N ) E. y e. ( ( G NeighbVtx N ) \ { x } ) p e. { <" x N y "> } <-> E. x e. ( G NeighbVtx N ) E. y e. ( ( G NeighbVtx N ) \ { x } ) z e. { <" x N y "> } ) )
109	106, 108	elab 3350	. . . . . . 7 |- ( z e. { p | E. x e. ( G NeighbVtx N ) E. y e. ( ( G NeighbVtx N ) \ { x } ) p e. { <" x N y "> } } <-> E. x e. ( G NeighbVtx N ) E. y e. ( ( G NeighbVtx N ) \ { x } ) z e. { <" x N y "> } )
110	109	bicomi 214	. . . . . 6 |- ( E. x e. ( G NeighbVtx N ) E. y e. ( ( G NeighbVtx N ) \ { x } ) z e. { <" x N y "> } <-> z e. { p | E. x e. ( G NeighbVtx N ) E. y e. ( ( G NeighbVtx N ) \ { x } ) p e. { <" x N y "> } } )
111	110	a1i 11	. . . . 5 |- ( ( G e. FinUSGraph /\ N e. V ) -> ( E. x e. ( G NeighbVtx N ) E. y e. ( ( G NeighbVtx N ) \ { x } ) z e. { <" x N y "> } <-> z e. { p | E. x e. ( G NeighbVtx N ) E. y e. ( ( G NeighbVtx N ) \ { x } ) p e. { <" x N y "> } } ) )
112	16, 105, 111	3bitrd 294	. . . 4 |- ( ( G e. FinUSGraph /\ N e. V ) -> ( ( z e. ( 2 WSPathsN G ) /\ ( z ` 1 ) = N ) <-> z e. { p | E. x e. ( G NeighbVtx N ) E. y e. ( ( G NeighbVtx N ) \ { x } ) p e. { <" x N y "> } } ) )
113	4, 112	bitrd 268	. . 3 |- ( ( G e. FinUSGraph /\ N e. V ) -> ( z e. ( M ` N ) <-> z e. { p | E. x e. ( G NeighbVtx N ) E. y e. ( ( G NeighbVtx N ) \ { x } ) p e. { <" x N y "> } } ) )
114	113	eqrdv 2620	. 2 |- ( ( G e. FinUSGraph /\ N e. V ) -> ( M ` N ) = { p | E. x e. ( G NeighbVtx N ) E. y e. ( ( G NeighbVtx N ) \ { x } ) p e. { <" x N y "> } } )
115		dfiunv2 4556	. 2 |- U_ x e. ( G NeighbVtx N ) U_ y e. ( ( G NeighbVtx N ) \ { x } ) { <" x N y "> } = { p | E. x e. ( G NeighbVtx N ) E. y e. ( ( G NeighbVtx N ) \ { x } ) p e. { <" x N y "> } }
116	114, 115	syl6eqr 2674	1 |- ( ( G e. FinUSGraph /\ N e. V ) -> ( M ` N ) = U_ x e. ( G NeighbVtx N ) U_ y e. ( ( G NeighbVtx N ) \ { x } ) { <" x N y "> } )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   =/= wne 2794   E.wrex 2913   {crab 2916   _Vcvv 3200   \ cdif 3571   {csn 4177   {cpr 4179   U_ciun 4520   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   1c1 9937   2c2 11070   NN0cn0 11292   <"cs3 13587  Vtxcvtx 25874  Edgcedg 25939   UMGraph cumgr 25976   USGraph cusgr 26044   FinUSGraph cfusgr 26208   NeighbVtx cnbgr 26224   WSPathsN cwwspthsn 26720   WSPathsNOn cwwspthsnon 26721

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-ac2 9285  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1013  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-ac 8939  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-s2 13593  df-s3 13594  df-edg 25940  df-uhgr 25953  df-upgr 25977  df-umgr 25978  df-uspgr 26045  df-usgr 26046  df-fusgr 26209  df-nbgr 26228  df-wlks 26495  df-wlkson 26496  df-trls 26589  df-trlson 26590  df-pths 26612  df-spths 26613  df-pthson 26614  df-spthson 26615  df-wwlks 26722  df-wwlksn 26723  df-wwlksnon 26724  df-wspthsn 26725  df-wspthsnon 26726

This theorem is referenced by:  fusgreghash2wspv  27199