Theorem extwwlkfab 27223

Description: The set ( X C N ) of closed walks (having a fixed length greater than one and starting at a fixed vertex) with the last but two vertex is identical with the first (and therefore last) vertex can be constructed from the set ( X F ( N - 2 ) ) of closed walks with length smaller by 2 than the fixed length appending a neighbor of the last vertex and afterwards the last vertex (which is the first vertex) itself ("walking forth and back" from the last vertex). 3 <_ N is required since for N = 2: ( X F ( N - 2 ) ) = ( X F 0 ) = (/), see umgrclwwlksge2 26912 stating that a closed walk of length 0 is not represented as word, at least not for an undirected simple graph. (Contributed by Alexander van der Vekens, 18-Sep-2018.) (Revised by AV, 29-May-2021.)

Hypotheses

Ref	Expression
extwwlkfab.v	|- V = (Vtx ` G )
extwwlkfab.f	|- F = ( v e. V , n e. NN |-> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } )
extwwlkfab.c	|- C = ( v e. V , n e. ( ZZ>= ` 2 ) |-> { w e. ( n ClWWalksN G ) | ( ( w ` 0 ) = v /\ ( w ` ( n - 2 ) ) = ( w ` 0 ) ) } )

Assertion

Ref	Expression
extwwlkfab	|- ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( X C N ) = { w e. ( N ClWWalksN G ) | ( ( w substr <. 0 , ( N - 2 ) >. ) e. ( X F ( N - 2 ) ) /\ ( w ` ( N - 1 ) ) e. ( G NeighbVtx X ) /\ ( w ` ( N - 2 ) ) = X ) } )

Distinct variable groups:   n, G, v, w   n, N, v, w   n, V, v, w   n, X, v, w

Allowed substitution hints:   C( w, v, n)   F( w, v, n)

Proof of Theorem extwwlkfab

Dummy variable u is distinct from all other variables.

Step	Hyp	Ref	Expression
1		uzuzle23 11729	. . . . 5 |- ( N e. ( ZZ>= ` 3 ) -> N e. ( ZZ>= ` 2 ) )
2	1	anim2i 593	. . . 4 |- ( ( X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( X e. V /\ N e. ( ZZ>= ` 2 ) ) )
3		extwwlkfab.c	. . . . 5 |- C = ( v e. V , n e. ( ZZ>= ` 2 ) |-> { w e. ( n ClWWalksN G ) | ( ( w ` 0 ) = v /\ ( w ` ( n - 2 ) ) = ( w ` 0 ) ) } )
4	3	numclwwlkovg 27220	. . . 4 |- ( ( X e. V /\ N e. ( ZZ>= ` 2 ) ) -> ( X C N ) = { w e. ( N ClWWalksN G ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) } )
5	2, 4	syl 17	. . 3 |- ( ( X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( X C N ) = { w e. ( N ClWWalksN G ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) } )
6	5	3adant1 1079	. 2 |- ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( X C N ) = { w e. ( N ClWWalksN G ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) } )
7		3simpb 1059	. . . . . . . . . 10 |- ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( G e. USGraph /\ N e. ( ZZ>= ` 3 ) ) )
8	7	adantr 481	. . . . . . . . 9 |- ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ w e. ( N ClWWalksN G ) ) -> ( G e. USGraph /\ N e. ( ZZ>= ` 3 ) ) )
9		simpr 477	. . . . . . . . 9 |- ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ w e. ( N ClWWalksN G ) ) -> w e. ( N ClWWalksN G ) )
10		simpr 477	. . . . . . . . 9 |- ( ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) -> ( w ` ( N - 2 ) ) = ( w ` 0 ) )
11		extwwlkfablem2 27210	. . . . . . . . 9 |- ( ( ( G e. USGraph /\ N e. ( ZZ>= ` 3 ) ) /\ w e. ( N ClWWalksN G ) /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) -> ( w substr <. 0 , ( N - 2 ) >. ) e. ( ( N - 2 ) ClWWalksN G ) )
12	8, 9, 10, 11	syl2an3an 1386	. . . . . . . 8 |- ( ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ w e. ( N ClWWalksN G ) ) /\ ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) ) -> ( w substr <. 0 , ( N - 2 ) >. ) e. ( ( N - 2 ) ClWWalksN G ) )
13		simpl 473	. . . . . . . . 9 |- ( ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) -> ( w ` 0 ) = X )
14	13	adantl 482	. . . . . . . 8 |- ( ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ w e. ( N ClWWalksN G ) ) /\ ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) ) -> ( w ` 0 ) = X )
15	12, 14	jca 554	. . . . . . 7 |- ( ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ w e. ( N ClWWalksN G ) ) /\ ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) ) -> ( ( w substr <. 0 , ( N - 2 ) >. ) e. ( ( N - 2 ) ClWWalksN G ) /\ ( w ` 0 ) = X ) )
16	1	anim2i 593	. . . . . . . . . . 11 |- ( ( G e. USGraph /\ N e. ( ZZ>= ` 3 ) ) -> ( G e. USGraph /\ N e. ( ZZ>= ` 2 ) ) )
17	16	3adant2 1080	. . . . . . . . . 10 |- ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( G e. USGraph /\ N e. ( ZZ>= ` 2 ) ) )
18	17	adantr 481	. . . . . . . . 9 |- ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ w e. ( N ClWWalksN G ) ) -> ( G e. USGraph /\ N e. ( ZZ>= ` 2 ) ) )
19		extwwlkfablem1 27207	. . . . . . . . 9 |- ( ( ( G e. USGraph /\ N e. ( ZZ>= ` 2 ) ) /\ w e. ( N ClWWalksN G ) /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) -> ( w ` ( N - 1 ) ) e. ( G NeighbVtx ( w ` 0 ) ) )
20	18, 9, 10, 19	syl2an3an 1386	. . . . . . . 8 |- ( ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ w e. ( N ClWWalksN G ) ) /\ ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) ) -> ( w ` ( N - 1 ) ) e. ( G NeighbVtx ( w ` 0 ) ) )
21		oveq2 6658	. . . . . . . . . . 11 |- ( X = ( w ` 0 ) -> ( G NeighbVtx X ) = ( G NeighbVtx ( w ` 0 ) ) )
22	21	eqcoms 2630	. . . . . . . . . 10 |- ( ( w ` 0 ) = X -> ( G NeighbVtx X ) = ( G NeighbVtx ( w ` 0 ) ) )
23	22	adantr 481	. . . . . . . . 9 |- ( ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) -> ( G NeighbVtx X ) = ( G NeighbVtx ( w ` 0 ) ) )
24	23	adantl 482	. . . . . . . 8 |- ( ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ w e. ( N ClWWalksN G ) ) /\ ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) ) -> ( G NeighbVtx X ) = ( G NeighbVtx ( w ` 0 ) ) )
25	20, 24	eleqtrrd 2704	. . . . . . 7 |- ( ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ w e. ( N ClWWalksN G ) ) /\ ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) ) -> ( w ` ( N - 1 ) ) e. ( G NeighbVtx X ) )
26	10, 13	eqtrd 2656	. . . . . . . 8 |- ( ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) -> ( w ` ( N - 2 ) ) = X )
27	26	adantl 482	. . . . . . 7 |- ( ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ w e. ( N ClWWalksN G ) ) /\ ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) ) -> ( w ` ( N - 2 ) ) = X )
28	15, 25, 27	3jca 1242	. . . . . 6 |- ( ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ w e. ( N ClWWalksN G ) ) /\ ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) ) -> ( ( ( w substr <. 0 , ( N - 2 ) >. ) e. ( ( N - 2 ) ClWWalksN G ) /\ ( w ` 0 ) = X ) /\ ( w ` ( N - 1 ) ) e. ( G NeighbVtx X ) /\ ( w ` ( N - 2 ) ) = X ) )
29	28	ex 450	. . . . 5 |- ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ w e. ( N ClWWalksN G ) ) -> ( ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) -> ( ( ( w substr <. 0 , ( N - 2 ) >. ) e. ( ( N - 2 ) ClWWalksN G ) /\ ( w ` 0 ) = X ) /\ ( w ` ( N - 1 ) ) e. ( G NeighbVtx X ) /\ ( w ` ( N - 2 ) ) = X ) ) )
30		simpl 473	. . . . . . . . . 10 |- ( ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = X ) -> ( w ` 0 ) = X )
31		simpr 477	. . . . . . . . . . 11 |- ( ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = X ) -> ( w ` ( N - 2 ) ) = X )
32	30	eqcomd 2628	. . . . . . . . . . 11 |- ( ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = X ) -> X = ( w ` 0 ) )
33	31, 32	eqtrd 2656	. . . . . . . . . 10 |- ( ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = X ) -> ( w ` ( N - 2 ) ) = ( w ` 0 ) )
34	30, 33	jca 554	. . . . . . . . 9 |- ( ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = X ) -> ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) )
35	34	ex 450	. . . . . . . 8 |- ( ( w ` 0 ) = X -> ( ( w ` ( N - 2 ) ) = X -> ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) ) )
36	35	a1d 25	. . . . . . 7 |- ( ( w ` 0 ) = X -> ( ( w ` ( N - 1 ) ) e. ( G NeighbVtx X ) -> ( ( w ` ( N - 2 ) ) = X -> ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) ) ) )
37	36	adantl 482	. . . . . 6 |- ( ( ( w substr <. 0 , ( N - 2 ) >. ) e. ( ( N - 2 ) ClWWalksN G ) /\ ( w ` 0 ) = X ) -> ( ( w ` ( N - 1 ) ) e. ( G NeighbVtx X ) -> ( ( w ` ( N - 2 ) ) = X -> ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) ) ) )
38	37	3imp 1256	. . . . 5 |- ( ( ( ( w substr <. 0 , ( N - 2 ) >. ) e. ( ( N - 2 ) ClWWalksN G ) /\ ( w ` 0 ) = X ) /\ ( w ` ( N - 1 ) ) e. ( G NeighbVtx X ) /\ ( w ` ( N - 2 ) ) = X ) -> ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) )
39	29, 38	impbid1 215	. . . 4 |- ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ w e. ( N ClWWalksN G ) ) -> ( ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) <-> ( ( ( w substr <. 0 , ( N - 2 ) >. ) e. ( ( N - 2 ) ClWWalksN G ) /\ ( w ` 0 ) = X ) /\ ( w ` ( N - 1 ) ) e. ( G NeighbVtx X ) /\ ( w ` ( N - 2 ) ) = X ) ) )
40		ige3m2fz 12365	. . . . . . . . . . . . . . 15 |- ( N e. ( ZZ>= ` 3 ) -> ( N - 2 ) e. ( 1 ... N ) )
41		oveq2 6658	. . . . . . . . . . . . . . . 16 |- ( ( # ` w ) = N -> ( 1 ... ( # ` w ) ) = ( 1 ... N ) )
42	41	eleq2d 2687	. . . . . . . . . . . . . . 15 |- ( ( # ` w ) = N -> ( ( N - 2 ) e. ( 1 ... ( # ` w ) ) <-> ( N - 2 ) e. ( 1 ... N ) ) )
43	40, 42	syl5ibr 236	. . . . . . . . . . . . . 14 |- ( ( # ` w ) = N -> ( N e. ( ZZ>= ` 3 ) -> ( N - 2 ) e. ( 1 ... ( # ` w ) ) ) )
44	43	adantl 482	. . . . . . . . . . . . 13 |- ( ( w e. Word V /\ ( # ` w ) = N ) -> ( N e. ( ZZ>= ` 3 ) -> ( N - 2 ) e. ( 1 ... ( # ` w ) ) ) )
45		simpl 473	. . . . . . . . . . . . 13 |- ( ( w e. Word V /\ ( # ` w ) = N ) -> w e. Word V )
46	44, 45	jctild 566	. . . . . . . . . . . 12 |- ( ( w e. Word V /\ ( # ` w ) = N ) -> ( N e. ( ZZ>= ` 3 ) -> ( w e. Word V /\ ( N - 2 ) e. ( 1 ... ( # ` w ) ) ) ) )
47		extwwlkfab.v	. . . . . . . . . . . . 13 |- V = (Vtx ` G )
48	47	clwwlknbp 26885	. . . . . . . . . . . 12 |- ( w e. ( N ClWWalksN G ) -> ( w e. Word V /\ ( # ` w ) = N ) )
49	46, 48	syl11 33	. . . . . . . . . . 11 |- ( N e. ( ZZ>= ` 3 ) -> ( w e. ( N ClWWalksN G ) -> ( w e. Word V /\ ( N - 2 ) e. ( 1 ... ( # ` w ) ) ) ) )
50	49	3ad2ant3 1084	. . . . . . . . . 10 |- ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( w e. ( N ClWWalksN G ) -> ( w e. Word V /\ ( N - 2 ) e. ( 1 ... ( # ` w ) ) ) ) )
51	50	imp 445	. . . . . . . . 9 |- ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ w e. ( N ClWWalksN G ) ) -> ( w e. Word V /\ ( N - 2 ) e. ( 1 ... ( # ` w ) ) ) )
52		swrd0fv0 13440	. . . . . . . . 9 |- ( ( w e. Word V /\ ( N - 2 ) e. ( 1 ... ( # ` w ) ) ) -> ( ( w substr <. 0 , ( N - 2 ) >. ) ` 0 ) = ( w ` 0 ) )
53	51, 52	syl 17	. . . . . . . 8 |- ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ w e. ( N ClWWalksN G ) ) -> ( ( w substr <. 0 , ( N - 2 ) >. ) ` 0 ) = ( w ` 0 ) )
54	53	eqcomd 2628	. . . . . . 7 |- ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ w e. ( N ClWWalksN G ) ) -> ( w ` 0 ) = ( ( w substr <. 0 , ( N - 2 ) >. ) ` 0 ) )
55	54	eqeq1d 2624	. . . . . 6 |- ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ w e. ( N ClWWalksN G ) ) -> ( ( w ` 0 ) = X <-> ( ( w substr <. 0 , ( N - 2 ) >. ) ` 0 ) = X ) )
56	55	anbi2d 740	. . . . 5 |- ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ w e. ( N ClWWalksN G ) ) -> ( ( ( w substr <. 0 , ( N - 2 ) >. ) e. ( ( N - 2 ) ClWWalksN G ) /\ ( w ` 0 ) = X ) <-> ( ( w substr <. 0 , ( N - 2 ) >. ) e. ( ( N - 2 ) ClWWalksN G ) /\ ( ( w substr <. 0 , ( N - 2 ) >. ) ` 0 ) = X ) ) )
57	56	3anbi1d 1403	. . . 4 |- ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ w e. ( N ClWWalksN G ) ) -> ( ( ( ( w substr <. 0 , ( N - 2 ) >. ) e. ( ( N - 2 ) ClWWalksN G ) /\ ( w ` 0 ) = X ) /\ ( w ` ( N - 1 ) ) e. ( G NeighbVtx X ) /\ ( w ` ( N - 2 ) ) = X ) <-> ( ( ( w substr <. 0 , ( N - 2 ) >. ) e. ( ( N - 2 ) ClWWalksN G ) /\ ( ( w substr <. 0 , ( N - 2 ) >. ) ` 0 ) = X ) /\ ( w ` ( N - 1 ) ) e. ( G NeighbVtx X ) /\ ( w ` ( N - 2 ) ) = X ) ) )
58		uz3m2nn 11731	. . . . . . . . . . 11 |- ( N e. ( ZZ>= ` 3 ) -> ( N - 2 ) e. NN )
59	58	anim2i 593	. . . . . . . . . 10 |- ( ( X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( X e. V /\ ( N - 2 ) e. NN ) )
60	59	3adant1 1079	. . . . . . . . 9 |- ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( X e. V /\ ( N - 2 ) e. NN ) )
61		extwwlkfab.f	. . . . . . . . . . 11 |- F = ( v e. V , n e. NN |-> { w e. ( n ClWWalksN G ) | ( w ` 0 ) = v } )
62	61	numclwwlkovf 27213	. . . . . . . . . 10 |- ( ( X e. V /\ ( N - 2 ) e. NN ) -> ( X F ( N - 2 ) ) = { w e. ( ( N - 2 ) ClWWalksN G ) | ( w ` 0 ) = X } )
63	62	eleq2d 2687	. . . . . . . . 9 |- ( ( X e. V /\ ( N - 2 ) e. NN ) -> ( ( w substr <. 0 , ( N - 2 ) >. ) e. ( X F ( N - 2 ) ) <-> ( w substr <. 0 , ( N - 2 ) >. ) e. { w e. ( ( N - 2 ) ClWWalksN G ) | ( w ` 0 ) = X } ) )
64	60, 63	syl 17	. . . . . . . 8 |- ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( ( w substr <. 0 , ( N - 2 ) >. ) e. ( X F ( N - 2 ) ) <-> ( w substr <. 0 , ( N - 2 ) >. ) e. { w e. ( ( N - 2 ) ClWWalksN G ) | ( w ` 0 ) = X } ) )
65		fveq1 6190	. . . . . . . . . 10 |- ( u = ( w substr <. 0 , ( N - 2 ) >. ) -> ( u ` 0 ) = ( ( w substr <. 0 , ( N - 2 ) >. ) ` 0 ) )
66	65	eqeq1d 2624	. . . . . . . . 9 |- ( u = ( w substr <. 0 , ( N - 2 ) >. ) -> ( ( u ` 0 ) = X <-> ( ( w substr <. 0 , ( N - 2 ) >. ) ` 0 ) = X ) )
67		fveq1 6190	. . . . . . . . . . 11 |- ( w = u -> ( w ` 0 ) = ( u ` 0 ) )
68	67	eqeq1d 2624	. . . . . . . . . 10 |- ( w = u -> ( ( w ` 0 ) = X <-> ( u ` 0 ) = X ) )
69	68	cbvrabv 3199	. . . . . . . . 9 |- { w e. ( ( N - 2 ) ClWWalksN G ) | ( w ` 0 ) = X } = { u e. ( ( N - 2 ) ClWWalksN G ) | ( u ` 0 ) = X }
70	66, 69	elrab2 3366	. . . . . . . 8 |- ( ( w substr <. 0 , ( N - 2 ) >. ) e. { w e. ( ( N - 2 ) ClWWalksN G ) | ( w ` 0 ) = X } <-> ( ( w substr <. 0 , ( N - 2 ) >. ) e. ( ( N - 2 ) ClWWalksN G ) /\ ( ( w substr <. 0 , ( N - 2 ) >. ) ` 0 ) = X ) )
71	64, 70	syl6bb 276	. . . . . . 7 |- ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( ( w substr <. 0 , ( N - 2 ) >. ) e. ( X F ( N - 2 ) ) <-> ( ( w substr <. 0 , ( N - 2 ) >. ) e. ( ( N - 2 ) ClWWalksN G ) /\ ( ( w substr <. 0 , ( N - 2 ) >. ) ` 0 ) = X ) ) )
72	71	3anbi1d 1403	. . . . . 6 |- ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( ( ( w substr <. 0 , ( N - 2 ) >. ) e. ( X F ( N - 2 ) ) /\ ( w ` ( N - 1 ) ) e. ( G NeighbVtx X ) /\ ( w ` ( N - 2 ) ) = X ) <-> ( ( ( w substr <. 0 , ( N - 2 ) >. ) e. ( ( N - 2 ) ClWWalksN G ) /\ ( ( w substr <. 0 , ( N - 2 ) >. ) ` 0 ) = X ) /\ ( w ` ( N - 1 ) ) e. ( G NeighbVtx X ) /\ ( w ` ( N - 2 ) ) = X ) ) )
73	72	bicomd 213	. . . . 5 |- ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( ( ( ( w substr <. 0 , ( N - 2 ) >. ) e. ( ( N - 2 ) ClWWalksN G ) /\ ( ( w substr <. 0 , ( N - 2 ) >. ) ` 0 ) = X ) /\ ( w ` ( N - 1 ) ) e. ( G NeighbVtx X ) /\ ( w ` ( N - 2 ) ) = X ) <-> ( ( w substr <. 0 , ( N - 2 ) >. ) e. ( X F ( N - 2 ) ) /\ ( w ` ( N - 1 ) ) e. ( G NeighbVtx X ) /\ ( w ` ( N - 2 ) ) = X ) ) )
74	73	adantr 481	. . . 4 |- ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ w e. ( N ClWWalksN G ) ) -> ( ( ( ( w substr <. 0 , ( N - 2 ) >. ) e. ( ( N - 2 ) ClWWalksN G ) /\ ( ( w substr <. 0 , ( N - 2 ) >. ) ` 0 ) = X ) /\ ( w ` ( N - 1 ) ) e. ( G NeighbVtx X ) /\ ( w ` ( N - 2 ) ) = X ) <-> ( ( w substr <. 0 , ( N - 2 ) >. ) e. ( X F ( N - 2 ) ) /\ ( w ` ( N - 1 ) ) e. ( G NeighbVtx X ) /\ ( w ` ( N - 2 ) ) = X ) ) )
75	39, 57, 74	3bitrd 294	. . 3 |- ( ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) /\ w e. ( N ClWWalksN G ) ) -> ( ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) <-> ( ( w substr <. 0 , ( N - 2 ) >. ) e. ( X F ( N - 2 ) ) /\ ( w ` ( N - 1 ) ) e. ( G NeighbVtx X ) /\ ( w ` ( N - 2 ) ) = X ) ) )
76	75	rabbidva 3188	. 2 |- ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> { w e. ( N ClWWalksN G ) | ( ( w ` 0 ) = X /\ ( w ` ( N - 2 ) ) = ( w ` 0 ) ) } = { w e. ( N ClWWalksN G ) | ( ( w substr <. 0 , ( N - 2 ) >. ) e. ( X F ( N - 2 ) ) /\ ( w ` ( N - 1 ) ) e. ( G NeighbVtx X ) /\ ( w ` ( N - 2 ) ) = X ) } )
77	6, 76	eqtrd 2656	1 |- ( ( G e. USGraph /\ X e. V /\ N e. ( ZZ>= ` 3 ) ) -> ( X C N ) = { w e. ( N ClWWalksN G ) | ( ( w substr <. 0 , ( N - 2 ) >. ) e. ( X F ( N - 2 ) ) /\ ( w ` ( N - 1 ) ) e. ( G NeighbVtx X ) /\ ( w ` ( N - 2 ) ) = X ) } )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {crab 2916   <.cop 4183   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   0cc0 9936   1c1 9937   - cmin 10266   NNcn 11020   2c2 11070   3c3 11071   ZZ>=cuz 11687   ...cfz 12326   #chash 13117  Word cword 13291   substr csubstr 13295  Vtxcvtx 25874   USGraph cusgr 26044   NeighbVtx cnbgr 26224   ClWWalksN cclwwlksn 26876

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-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-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-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-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-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-lsw 13300  df-substr 13303  df-edg 25940  df-upgr 25977  df-umgr 25978  df-usgr 26046  df-nbgr 26228  df-wwlks 26722  df-wwlksn 26723  df-clwwlks 26877  df-clwwlksn 26878

This theorem is referenced by:  numclwlk1lem2foa  27224  numclwlk1lem2f  27225