Theorem wlkres 16259

Description: The restriction <. H , Q >. of a walk <. F , P >. to an initial segment of the walk (of length N) forms a walk on the subgraph S consisting of the edges in the initial segment. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.) (Revised by AV, 5-Mar-2021.) Hypothesis revised using the prefix operation. (Revised by AV, 30-Nov-2022.)

Hypotheses

Ref	Expression
wlkres.v	|- V = (Vtx ` G )
wlkres.i	|- I = (iEdg ` G )
wlkres.d	|- ( ph -> F (Walks ` G ) P )
wlkres.n	|- ( ph -> N e. ( 0..^ (♯ ` F ) ) )
wlkres.s	|- ( ph -> (Vtx ` S ) = V )
wlkres.e	|- ( ph -> (iEdg ` S ) = ( I |` ( F " ( 0..^ N ) ) ) )
wlkres.h	|- H = ( F prefix N )
wlkres.q	|- Q = ( P |` ( 0 ... N ) )

Assertion

Ref	Expression
wlkres	|- ( ph -> H (Walks ` S ) Q )

Proof of Theorem wlkres

Dummy variables x k are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		wlkres.d	. . . . 5 |- ( ph -> F (Walks ` G ) P )
2		wlkres.i	. . . . . 6 |- I = (iEdg ` G )
3	2	wlkf 16210	. . . . 5 |- ( F (Walks ` G ) P -> F e. Word dom I )
4	1, 3	syl 14	. . . 4 |- ( ph -> F e. Word dom I )
5		wlkres.n	. . . . 5 |- ( ph -> N e. ( 0..^ (♯ ` F ) ) )
6		elfzonn0 10431	. . . . 5 |- ( N e. ( 0..^ (♯ ` F ) ) -> N e. NN0 )
7	5, 6	syl 14	. . . 4 |- ( ph -> N e. NN0 )
8		pfxwrdsymbg 11280	. . . 4 |- ( ( F e. Word dom I /\ N e. NN0 ) -> ( F prefix N ) e. Word ( F " ( 0..^ N ) ) )
9	4, 7, 8	syl2anc 411	. . 3 |- ( ph -> ( F prefix N ) e. Word ( F " ( 0..^ N ) ) )
10		wlkres.h	. . . 4 |- H = ( F prefix N )
11	10	a1i 9	. . 3 |- ( ph -> H = ( F prefix N ) )
12		wlkres.e	. . . . . 6 |- ( ph -> (iEdg ` S ) = ( I |` ( F " ( 0..^ N ) ) ) )
13	12	dmeqd 4935	. . . . 5 |- ( ph -> dom (iEdg ` S ) = dom ( I |` ( F " ( 0..^ N ) ) ) )
14		wrdf 11128	. . . . . . 7 |- ( F e. Word dom I -> F : ( 0..^ (♯ ` F ) ) --> dom I )
15		fimass 5500	. . . . . . 7 |- ( F : ( 0..^ (♯ ` F ) ) --> dom I -> ( F " ( 0..^ N ) ) C_ dom I )
16	4, 14, 15	3syl 17	. . . . . 6 |- ( ph -> ( F " ( 0..^ N ) ) C_ dom I )
17		ssdmres 5037	. . . . . 6 |- ( ( F " ( 0..^ N ) ) C_ dom I <-> dom ( I |` ( F " ( 0..^ N ) ) ) = ( F " ( 0..^ N ) ) )
18	16, 17	sylib 122	. . . . 5 |- ( ph -> dom ( I |` ( F " ( 0..^ N ) ) ) = ( F " ( 0..^ N ) ) )
19	13, 18	eqtrd 2263	. . . 4 |- ( ph -> dom (iEdg ` S ) = ( F " ( 0..^ N ) ) )
20		wrdeq 11144	. . . 4 |- ( dom (iEdg ` S ) = ( F " ( 0..^ N ) ) -> Word dom (iEdg ` S ) = Word ( F " ( 0..^ N ) ) )
21	19, 20	syl 14	. . 3 |- ( ph -> Word dom (iEdg ` S ) = Word ( F " ( 0..^ N ) ) )
22	9, 11, 21	3eltr4d 2314	. 2 |- ( ph -> H e. Word dom (iEdg ` S ) )
23		wlkres.v	. . . . . . . 8 |- V = (Vtx ` G )
24	23	wlkp 16214	. . . . . . 7 |- ( F (Walks ` G ) P -> P : ( 0 ... (♯ ` F ) ) --> V )
25	1, 24	syl 14	. . . . . 6 |- ( ph -> P : ( 0 ... (♯ ` F ) ) --> V )
26		wlkres.s	. . . . . . 7 |- ( ph -> (Vtx ` S ) = V )
27	26	feq3d 5473	. . . . . 6 |- ( ph -> ( P : ( 0 ... (♯ ` F ) ) --> (Vtx ` S ) <-> P : ( 0 ... (♯ ` F ) ) --> V ) )
28	25, 27	mpbird 167	. . . . 5 |- ( ph -> P : ( 0 ... (♯ ` F ) ) --> (Vtx ` S ) )
29		fzossfz 10406	. . . . . . 7 |- ( 0..^ (♯ ` F ) ) C_ ( 0 ... (♯ ` F ) )
30	29, 5	sselid 3224	. . . . . 6 |- ( ph -> N e. ( 0 ... (♯ ` F ) ) )
31		elfzuz3 10262	. . . . . 6 |- ( N e. ( 0 ... (♯ ` F ) ) -> (♯ ` F ) e. ( ZZ>= ` N ) )
32		fzss2 10304	. . . . . 6 |- ( (♯ ` F ) e. ( ZZ>= ` N ) -> ( 0 ... N ) C_ ( 0 ... (♯ ` F ) ) )
33	30, 31, 32	3syl 17	. . . . 5 |- ( ph -> ( 0 ... N ) C_ ( 0 ... (♯ ` F ) ) )
34	28, 33	fssresd 5515	. . . 4 |- ( ph -> ( P |` ( 0 ... N ) ) : ( 0 ... N ) --> (Vtx ` S ) )
35	10	fveq2i 5645	. . . . . . 7 |- (♯ ` H ) = (♯ ` ( F prefix N ) )
36		pfxlen 11275	. . . . . . . 8 |- ( ( F e. Word dom I /\ N e. ( 0 ... (♯ ` F ) ) ) -> (♯ ` ( F prefix N ) ) = N )
37	4, 30, 36	syl2anc 411	. . . . . . 7 |- ( ph -> (♯ ` ( F prefix N ) ) = N )
38	35, 37	eqtrid 2275	. . . . . 6 |- ( ph -> (♯ ` H ) = N )
39	38	oveq2d 6039	. . . . 5 |- ( ph -> ( 0 ... (♯ ` H ) ) = ( 0 ... N ) )
40	39	feq2d 5472	. . . 4 |- ( ph -> ( ( P |` ( 0 ... N ) ) : ( 0 ... (♯ ` H ) ) --> (Vtx ` S ) <-> ( P |` ( 0 ... N ) ) : ( 0 ... N ) --> (Vtx ` S ) ) )
41	34, 40	mpbird 167	. . 3 |- ( ph -> ( P |` ( 0 ... N ) ) : ( 0 ... (♯ ` H ) ) --> (Vtx ` S ) )
42		wlkres.q	. . . 4 |- Q = ( P |` ( 0 ... N ) )
43	42	feq1i 5477	. . 3 |- ( Q : ( 0 ... (♯ ` H ) ) --> (Vtx ` S ) <-> ( P |` ( 0 ... N ) ) : ( 0 ... (♯ ` H ) ) --> (Vtx ` S ) )
44	41, 43	sylibr 134	. 2 |- ( ph -> Q : ( 0 ... (♯ ` H ) ) --> (Vtx ` S ) )
45	23, 2	wlkprop 16207	. . . . . 6 |- ( F (Walks ` G ) P -> ( F e. Word dom I /\ P : ( 0 ... (♯ ` F ) ) --> V /\ A. k e. ( 0..^ (♯ ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) ) )
46	1, 45	syl 14	. . . . 5 |- ( ph -> ( F e. Word dom I /\ P : ( 0 ... (♯ ` F ) ) --> V /\ A. k e. ( 0..^ (♯ ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) ) )
47	46	adantr 276	. . . 4 |- ( ( ph /\ x e. ( 0..^ (♯ ` H ) ) ) -> ( F e. Word dom I /\ P : ( 0 ... (♯ ` F ) ) --> V /\ A. k e. ( 0..^ (♯ ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) ) )
48	38	oveq2d 6039	. . . . . . . . . . 11 |- ( ph -> ( 0..^ (♯ ` H ) ) = ( 0..^ N ) )
49	48	eleq2d 2300	. . . . . . . . . 10 |- ( ph -> ( x e. ( 0..^ (♯ ` H ) ) <-> x e. ( 0..^ N ) ) )
50	42	fveq1i 5643	. . . . . . . . . . . . 13 |- ( Q ` x ) = ( ( P |` ( 0 ... N ) ) ` x )
51		fzossfz 10406	. . . . . . . . . . . . . . . 16 |- ( 0..^ N ) C_ ( 0 ... N )
52	51	a1i 9	. . . . . . . . . . . . . . 15 |- ( ph -> ( 0..^ N ) C_ ( 0 ... N ) )
53	52	sselda 3226	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. ( 0..^ N ) ) -> x e. ( 0 ... N ) )
54	53	fvresd 5667	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. ( 0..^ N ) ) -> ( ( P |` ( 0 ... N ) ) ` x ) = ( P ` x ) )
55	50, 54	eqtr2id 2276	. . . . . . . . . . . 12 |- ( ( ph /\ x e. ( 0..^ N ) ) -> ( P ` x ) = ( Q ` x ) )
56	42	fveq1i 5643	. . . . . . . . . . . . 13 |- ( Q ` ( x + 1 ) ) = ( ( P |` ( 0 ... N ) ) ` ( x + 1 ) )
57		fzofzp1 10478	. . . . . . . . . . . . . . 15 |- ( x e. ( 0..^ N ) -> ( x + 1 ) e. ( 0 ... N ) )
58	57	adantl 277	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. ( 0..^ N ) ) -> ( x + 1 ) e. ( 0 ... N ) )
59	58	fvresd 5667	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. ( 0..^ N ) ) -> ( ( P |` ( 0 ... N ) ) ` ( x + 1 ) ) = ( P ` ( x + 1 ) ) )
60	56, 59	eqtr2id 2276	. . . . . . . . . . . 12 |- ( ( ph /\ x e. ( 0..^ N ) ) -> ( P ` ( x + 1 ) ) = ( Q ` ( x + 1 ) ) )
61	55, 60	jca 306	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( 0..^ N ) ) -> ( ( P ` x ) = ( Q ` x ) /\ ( P ` ( x + 1 ) ) = ( Q ` ( x + 1 ) ) ) )
62	61	ex 115	. . . . . . . . . 10 |- ( ph -> ( x e. ( 0..^ N ) -> ( ( P ` x ) = ( Q ` x ) /\ ( P ` ( x + 1 ) ) = ( Q ` ( x + 1 ) ) ) ) )
63	49, 62	sylbid 150	. . . . . . . . 9 |- ( ph -> ( x e. ( 0..^ (♯ ` H ) ) -> ( ( P ` x ) = ( Q ` x ) /\ ( P ` ( x + 1 ) ) = ( Q ` ( x + 1 ) ) ) ) )
64	63	imp 124	. . . . . . . 8 |- ( ( ph /\ x e. ( 0..^ (♯ ` H ) ) ) -> ( ( P ` x ) = ( Q ` x ) /\ ( P ` ( x + 1 ) ) = ( Q ` ( x + 1 ) ) ) )
65	4	ancli 323	. . . . . . . . . . . . . 14 |- ( ph -> ( ph /\ F e. Word dom I ) )
66	14	ffund 5488	. . . . . . . . . . . . . . . . 17 |- ( F e. Word dom I -> Fun F )
67	66	adantl 277	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ F e. Word dom I ) -> Fun F )
68	67	adantr 276	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ F e. Word dom I ) /\ x e. ( 0..^ N ) ) -> Fun F )
69		fdm 5490	. . . . . . . . . . . . . . . . . 18 |- ( F : ( 0..^ (♯ ` F ) ) --> dom I -> dom F = ( 0..^ (♯ ` F ) ) )
70		elfzouz2 10402	. . . . . . . . . . . . . . . . . . . 20 |- ( N e. ( 0..^ (♯ ` F ) ) -> (♯ ` F ) e. ( ZZ>= ` N ) )
71		fzoss2 10414	. . . . . . . . . . . . . . . . . . . 20 |- ( (♯ ` F ) e. ( ZZ>= ` N ) -> ( 0..^ N ) C_ ( 0..^ (♯ ` F ) ) )
72	5, 70, 71	3syl 17	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> ( 0..^ N ) C_ ( 0..^ (♯ ` F ) ) )
73		sseq2 3250	. . . . . . . . . . . . . . . . . . 19 |- ( dom F = ( 0..^ (♯ ` F ) ) -> ( ( 0..^ N ) C_ dom F <-> ( 0..^ N ) C_ ( 0..^ (♯ ` F ) ) ) )
74	72, 73	imbitrrid 156	. . . . . . . . . . . . . . . . . 18 |- ( dom F = ( 0..^ (♯ ` F ) ) -> ( ph -> ( 0..^ N ) C_ dom F ) )
75	14, 69, 74	3syl 17	. . . . . . . . . . . . . . . . 17 |- ( F e. Word dom I -> ( ph -> ( 0..^ N ) C_ dom F ) )
76	75	impcom 125	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ F e. Word dom I ) -> ( 0..^ N ) C_ dom F )
77	76	adantr 276	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ F e. Word dom I ) /\ x e. ( 0..^ N ) ) -> ( 0..^ N ) C_ dom F )
78		simpr 110	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ F e. Word dom I ) /\ x e. ( 0..^ N ) ) -> x e. ( 0..^ N ) )
79	68, 77, 78	resfvresima 5896	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ F e. Word dom I ) /\ x e. ( 0..^ N ) ) -> ( ( I |` ( F " ( 0..^ N ) ) ) ` ( ( F |` ( 0..^ N ) ) ` x ) ) = ( I ` ( F ` x ) ) )
80	65, 79	sylan 283	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. ( 0..^ N ) ) -> ( ( I |` ( F " ( 0..^ N ) ) ) ` ( ( F |` ( 0..^ N ) ) ` x ) ) = ( I ` ( F ` x ) ) )
81	80	eqcomd 2236	. . . . . . . . . . . 12 |- ( ( ph /\ x e. ( 0..^ N ) ) -> ( I ` ( F ` x ) ) = ( ( I |` ( F " ( 0..^ N ) ) ) ` ( ( F |` ( 0..^ N ) ) ` x ) ) )
82	81	ex 115	. . . . . . . . . . 11 |- ( ph -> ( x e. ( 0..^ N ) -> ( I ` ( F ` x ) ) = ( ( I |` ( F " ( 0..^ N ) ) ) ` ( ( F |` ( 0..^ N ) ) ` x ) ) ) )
83	49, 82	sylbid 150	. . . . . . . . . 10 |- ( ph -> ( x e. ( 0..^ (♯ ` H ) ) -> ( I ` ( F ` x ) ) = ( ( I |` ( F " ( 0..^ N ) ) ) ` ( ( F |` ( 0..^ N ) ) ` x ) ) ) )
84	83	imp 124	. . . . . . . . 9 |- ( ( ph /\ x e. ( 0..^ (♯ ` H ) ) ) -> ( I ` ( F ` x ) ) = ( ( I |` ( F " ( 0..^ N ) ) ) ` ( ( F |` ( 0..^ N ) ) ` x ) ) )
85	12	adantr 276	. . . . . . . . . 10 |- ( ( ph /\ x e. ( 0..^ (♯ ` H ) ) ) -> (iEdg ` S ) = ( I |` ( F " ( 0..^ N ) ) ) )
86	10	fveq1i 5643	. . . . . . . . . . 11 |- ( H ` x ) = ( ( F prefix N ) ` x )
87	4	adantr 276	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. ( 0..^ (♯ ` H ) ) ) -> F e. Word dom I )
88	30	adantr 276	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. ( 0..^ (♯ ` H ) ) ) -> N e. ( 0 ... (♯ ` F ) ) )
89		pfxres 11271	. . . . . . . . . . . . 13 |- ( ( F e. Word dom I /\ N e. ( 0 ... (♯ ` F ) ) ) -> ( F prefix N ) = ( F |` ( 0..^ N ) ) )
90	87, 88, 89	syl2anc 411	. . . . . . . . . . . 12 |- ( ( ph /\ x e. ( 0..^ (♯ ` H ) ) ) -> ( F prefix N ) = ( F |` ( 0..^ N ) ) )
91	90	fveq1d 5644	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( 0..^ (♯ ` H ) ) ) -> ( ( F prefix N ) ` x ) = ( ( F |` ( 0..^ N ) ) ` x ) )
92	86, 91	eqtrid 2275	. . . . . . . . . 10 |- ( ( ph /\ x e. ( 0..^ (♯ ` H ) ) ) -> ( H ` x ) = ( ( F |` ( 0..^ N ) ) ` x ) )
93	85, 92	fveq12d 5649	. . . . . . . . 9 |- ( ( ph /\ x e. ( 0..^ (♯ ` H ) ) ) -> ( (iEdg ` S ) ` ( H ` x ) ) = ( ( I |` ( F " ( 0..^ N ) ) ) ` ( ( F |` ( 0..^ N ) ) ` x ) ) )
94	84, 93	eqtr4d 2266	. . . . . . . 8 |- ( ( ph /\ x e. ( 0..^ (♯ ` H ) ) ) -> ( I ` ( F ` x ) ) = ( (iEdg ` S ) ` ( H ` x ) ) )
95	64, 94	jca 306	. . . . . . 7 |- ( ( ph /\ x e. ( 0..^ (♯ ` H ) ) ) -> ( ( ( P ` x ) = ( Q ` x ) /\ ( P ` ( x + 1 ) ) = ( Q ` ( x + 1 ) ) ) /\ ( I ` ( F ` x ) ) = ( (iEdg ` S ) ` ( H ` x ) ) ) )
96	5, 70	syl 14	. . . . . . . . . . 11 |- ( ph -> (♯ ` F ) e. ( ZZ>= ` N ) )
97	38	fveq2d 5646	. . . . . . . . . . 11 |- ( ph -> ( ZZ>= ` (♯ ` H ) ) = ( ZZ>= ` N ) )
98	96, 97	eleqtrrd 2310	. . . . . . . . . 10 |- ( ph -> (♯ ` F ) e. ( ZZ>= ` (♯ ` H ) ) )
99		fzoss2 10414	. . . . . . . . . 10 |- ( (♯ ` F ) e. ( ZZ>= ` (♯ ` H ) ) -> ( 0..^ (♯ ` H ) ) C_ ( 0..^ (♯ ` F ) ) )
100	98, 99	syl 14	. . . . . . . . 9 |- ( ph -> ( 0..^ (♯ ` H ) ) C_ ( 0..^ (♯ ` F ) ) )
101	100	sselda 3226	. . . . . . . 8 |- ( ( ph /\ x e. ( 0..^ (♯ ` H ) ) ) -> x e. ( 0..^ (♯ ` F ) ) )
102		wkslem1 16200	. . . . . . . . 9 |- ( k = x -> (if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) <-> if- ( ( P ` x ) = ( P ` ( x + 1 ) ) , ( I ` ( F ` x ) ) = { ( P ` x ) } , { ( P ` x ) , ( P ` ( x + 1 ) ) } C_ ( I ` ( F ` x ) ) ) ) )
103	102	rspcv 2905	. . . . . . . 8 |- ( x e. ( 0..^ (♯ ` F ) ) -> ( A. k e. ( 0..^ (♯ ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) -> if- ( ( P ` x ) = ( P ` ( x + 1 ) ) , ( I ` ( F ` x ) ) = { ( P ` x ) } , { ( P ` x ) , ( P ` ( x + 1 ) ) } C_ ( I ` ( F ` x ) ) ) ) )
104	101, 103	syl 14	. . . . . . 7 |- ( ( ph /\ x e. ( 0..^ (♯ ` H ) ) ) -> ( A. k e. ( 0..^ (♯ ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) -> if- ( ( P ` x ) = ( P ` ( x + 1 ) ) , ( I ` ( F ` x ) ) = { ( P ` x ) } , { ( P ` x ) , ( P ` ( x + 1 ) ) } C_ ( I ` ( F ` x ) ) ) ) )
105		eqeq12 2243	. . . . . . . . . 10 |- ( ( ( P ` x ) = ( Q ` x ) /\ ( P ` ( x + 1 ) ) = ( Q ` ( x + 1 ) ) ) -> ( ( P ` x ) = ( P ` ( x + 1 ) ) <-> ( Q ` x ) = ( Q ` ( x + 1 ) ) ) )
106	105	adantr 276	. . . . . . . . 9 |- ( ( ( ( P ` x ) = ( Q ` x ) /\ ( P ` ( x + 1 ) ) = ( Q ` ( x + 1 ) ) ) /\ ( I ` ( F ` x ) ) = ( (iEdg ` S ) ` ( H ` x ) ) ) -> ( ( P ` x ) = ( P ` ( x + 1 ) ) <-> ( Q ` x ) = ( Q ` ( x + 1 ) ) ) )
107		simpr 110	. . . . . . . . . 10 |- ( ( ( ( P ` x ) = ( Q ` x ) /\ ( P ` ( x + 1 ) ) = ( Q ` ( x + 1 ) ) ) /\ ( I ` ( F ` x ) ) = ( (iEdg ` S ) ` ( H ` x ) ) ) -> ( I ` ( F ` x ) ) = ( (iEdg ` S ) ` ( H ` x ) ) )
108		sneq 3681	. . . . . . . . . . . 12 |- ( ( P ` x ) = ( Q ` x ) -> { ( P ` x ) } = { ( Q ` x ) } )
109	108	adantr 276	. . . . . . . . . . 11 |- ( ( ( P ` x ) = ( Q ` x ) /\ ( P ` ( x + 1 ) ) = ( Q ` ( x + 1 ) ) ) -> { ( P ` x ) } = { ( Q ` x ) } )
110	109	adantr 276	. . . . . . . . . 10 |- ( ( ( ( P ` x ) = ( Q ` x ) /\ ( P ` ( x + 1 ) ) = ( Q ` ( x + 1 ) ) ) /\ ( I ` ( F ` x ) ) = ( (iEdg ` S ) ` ( H ` x ) ) ) -> { ( P ` x ) } = { ( Q ` x ) } )
111	107, 110	eqeq12d 2245	. . . . . . . . 9 |- ( ( ( ( P ` x ) = ( Q ` x ) /\ ( P ` ( x + 1 ) ) = ( Q ` ( x + 1 ) ) ) /\ ( I ` ( F ` x ) ) = ( (iEdg ` S ) ` ( H ` x ) ) ) -> ( ( I ` ( F ` x ) ) = { ( P ` x ) } <-> ( (iEdg ` S ) ` ( H ` x ) ) = { ( Q ` x ) } ) )
112		preq12 3751	. . . . . . . . . . 11 |- ( ( ( P ` x ) = ( Q ` x ) /\ ( P ` ( x + 1 ) ) = ( Q ` ( x + 1 ) ) ) -> { ( P ` x ) , ( P ` ( x + 1 ) ) } = { ( Q ` x ) , ( Q ` ( x + 1 ) ) } )
113	112	adantr 276	. . . . . . . . . 10 |- ( ( ( ( P ` x ) = ( Q ` x ) /\ ( P ` ( x + 1 ) ) = ( Q ` ( x + 1 ) ) ) /\ ( I ` ( F ` x ) ) = ( (iEdg ` S ) ` ( H ` x ) ) ) -> { ( P ` x ) , ( P ` ( x + 1 ) ) } = { ( Q ` x ) , ( Q ` ( x + 1 ) ) } )
114	113, 107	sseq12d 3257	. . . . . . . . 9 |- ( ( ( ( P ` x ) = ( Q ` x ) /\ ( P ` ( x + 1 ) ) = ( Q ` ( x + 1 ) ) ) /\ ( I ` ( F ` x ) ) = ( (iEdg ` S ) ` ( H ` x ) ) ) -> ( { ( P ` x ) , ( P ` ( x + 1 ) ) } C_ ( I ` ( F ` x ) ) <-> { ( Q ` x ) , ( Q ` ( x + 1 ) ) } C_ ( (iEdg ` S ) ` ( H ` x ) ) ) )
115	106, 111, 114	ifpbi123d 1000	. . . . . . . 8 |- ( ( ( ( P ` x ) = ( Q ` x ) /\ ( P ` ( x + 1 ) ) = ( Q ` ( x + 1 ) ) ) /\ ( I ` ( F ` x ) ) = ( (iEdg ` S ) ` ( H ` x ) ) ) -> (if- ( ( P ` x ) = ( P ` ( x + 1 ) ) , ( I ` ( F ` x ) ) = { ( P ` x ) } , { ( P ` x ) , ( P ` ( x + 1 ) ) } C_ ( I ` ( F ` x ) ) ) <-> if- ( ( Q ` x ) = ( Q ` ( x + 1 ) ) , ( (iEdg ` S ) ` ( H ` x ) ) = { ( Q ` x ) } , { ( Q ` x ) , ( Q ` ( x + 1 ) ) } C_ ( (iEdg ` S ) ` ( H ` x ) ) ) ) )
116	115	biimpd 144	. . . . . . 7 |- ( ( ( ( P ` x ) = ( Q ` x ) /\ ( P ` ( x + 1 ) ) = ( Q ` ( x + 1 ) ) ) /\ ( I ` ( F ` x ) ) = ( (iEdg ` S ) ` ( H ` x ) ) ) -> (if- ( ( P ` x ) = ( P ` ( x + 1 ) ) , ( I ` ( F ` x ) ) = { ( P ` x ) } , { ( P ` x ) , ( P ` ( x + 1 ) ) } C_ ( I ` ( F ` x ) ) ) -> if- ( ( Q ` x ) = ( Q ` ( x + 1 ) ) , ( (iEdg ` S ) ` ( H ` x ) ) = { ( Q ` x ) } , { ( Q ` x ) , ( Q ` ( x + 1 ) ) } C_ ( (iEdg ` S ) ` ( H ` x ) ) ) ) )
117	95, 104, 116	sylsyld 58	. . . . . 6 |- ( ( ph /\ x e. ( 0..^ (♯ ` H ) ) ) -> ( A. k e. ( 0..^ (♯ ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) -> if- ( ( Q ` x ) = ( Q ` ( x + 1 ) ) , ( (iEdg ` S ) ` ( H ` x ) ) = { ( Q ` x ) } , { ( Q ` x ) , ( Q ` ( x + 1 ) ) } C_ ( (iEdg ` S ) ` ( H ` x ) ) ) ) )
118	117	com12 30	. . . . 5 |- ( A. k e. ( 0..^ (♯ ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) -> ( ( ph /\ x e. ( 0..^ (♯ ` H ) ) ) -> if- ( ( Q ` x ) = ( Q ` ( x + 1 ) ) , ( (iEdg ` S ) ` ( H ` x ) ) = { ( Q ` x ) } , { ( Q ` x ) , ( Q ` ( x + 1 ) ) } C_ ( (iEdg ` S ) ` ( H ` x ) ) ) ) )
119	118	3ad2ant3 1046	. . . 4 |- ( ( F e. Word dom I /\ P : ( 0 ... (♯ ` F ) ) --> V /\ A. k e. ( 0..^ (♯ ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) ) -> ( ( ph /\ x e. ( 0..^ (♯ ` H ) ) ) -> if- ( ( Q ` x ) = ( Q ` ( x + 1 ) ) , ( (iEdg ` S ) ` ( H ` x ) ) = { ( Q ` x ) } , { ( Q ` x ) , ( Q ` ( x + 1 ) ) } C_ ( (iEdg ` S ) ` ( H ` x ) ) ) ) )
120	47, 119	mpcom 36	. . 3 |- ( ( ph /\ x e. ( 0..^ (♯ ` H ) ) ) -> if- ( ( Q ` x ) = ( Q ` ( x + 1 ) ) , ( (iEdg ` S ) ` ( H ` x ) ) = { ( Q ` x ) } , { ( Q ` x ) , ( Q ` ( x + 1 ) ) } C_ ( (iEdg ` S ) ` ( H ` x ) ) ) )
121	120	ralrimiva 2604	. 2 |- ( ph -> A. x e. ( 0..^ (♯ ` H ) )if- ( ( Q ` x ) = ( Q ` ( x + 1 ) ) , ( (iEdg ` S ) ` ( H ` x ) ) = { ( Q ` x ) } , { ( Q ` x ) , ( Q ` ( x + 1 ) ) } C_ ( (iEdg ` S ) ` ( H ` x ) ) ) )
122	23, 2, 1, 5, 26	wlkreslem 16258	. . 3 |- ( ph -> S e. _V )
123		eqid 2230	. . . 4 |- (Vtx ` S ) = (Vtx ` S )
124		eqid 2230	. . . 4 |- (iEdg ` S ) = (iEdg ` S )
125	123, 124	iswlkg 16209	. . 3 |- ( S e. _V -> ( H (Walks ` S ) Q <-> ( H e. Word dom (iEdg ` S ) /\ Q : ( 0 ... (♯ ` H ) ) --> (Vtx ` S ) /\ A. x e. ( 0..^ (♯ ` H ) )if- ( ( Q ` x ) = ( Q ` ( x + 1 ) ) , ( (iEdg ` S ) ` ( H ` x ) ) = { ( Q ` x ) } , { ( Q ` x ) , ( Q ` ( x + 1 ) ) } C_ ( (iEdg ` S ) ` ( H ` x ) ) ) ) ) )
126	122, 125	syl 14	. 2 |- ( ph -> ( H (Walks ` S ) Q <-> ( H e. Word dom (iEdg ` S ) /\ Q : ( 0 ... (♯ ` H ) ) --> (Vtx ` S ) /\ A. x e. ( 0..^ (♯ ` H ) )if- ( ( Q ` x ) = ( Q ` ( x + 1 ) ) , ( (iEdg ` S ) ` ( H ` x ) ) = { ( Q ` x ) } , { ( Q ` x ) , ( Q ` ( x + 1 ) ) } C_ ( (iEdg ` S ) ` ( H ` x ) ) ) ) ) )
127	22, 44, 121, 126	mpbir3and 1206	1 |- ( ph -> H (Walks ` S ) Q )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105  if-wif 985   /\ w3a 1004   = wceq 1397   e. wcel 2201   A.wral 2509   _Vcvv 2801   C_ wss 3199   {csn 3670   {cpr 3671   class class class wbr 4089   dom cdm 4727   |` cres 4729   "cima 4730   Fun wfun 5322   -->wf 5324   ` cfv 5328  (class class class)co 6023   0cc0 8037   1c1 8038   + caddc 8040   NN0cn0 9407   ZZ>=cuz 9760   ...cfz 10248  ..^cfzo 10382  ♯chash 11043  Word cword 11122   prefix cpfx 11262  Vtxcvtx 15892  iEdgciedg 15893  Walkscwlks 16197

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-coll 4205  ax-sep 4208  ax-nul 4216  ax-pow 4266  ax-pr 4301  ax-un 4532  ax-setind 4637  ax-iinf 4688  ax-cnex 8128  ax-resscn 8129  ax-1cn 8130  ax-1re 8131  ax-icn 8132  ax-addcl 8133  ax-addrcl 8134  ax-mulcl 8135  ax-addcom 8137  ax-mulcom 8138  ax-addass 8139  ax-mulass 8140  ax-distr 8141  ax-i2m1 8142  ax-0lt1 8143  ax-1rid 8144  ax-0id 8145  ax-rnegex 8146  ax-cnre 8148  ax-pre-ltirr 8149  ax-pre-ltwlin 8150  ax-pre-lttrn 8151  ax-pre-apti 8152  ax-pre-ltadd 8153

This theorem depends on definitions:  df-bi 117  df-dc 842  df-ifp 986  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-nel 2497  df-ral 2514  df-rex 2515  df-reu 2516  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-nul 3494  df-if 3605  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-int 3930  df-iun 3973  df-br 4090  df-opab 4152  df-mpt 4153  df-tr 4189  df-id 4392  df-iord 4465  df-on 4467  df-ilim 4468  df-suc 4470  df-iom 4691  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-ima 4740  df-iota 5288  df-fun 5330  df-fn 5331  df-f 5332  df-f1 5333  df-fo 5334  df-f1o 5335  df-fv 5336  df-riota 5976  df-ov 6026  df-oprab 6027  df-mpo 6028  df-1st 6308  df-2nd 6309  df-recs 6476  df-frec 6562  df-1o 6587  df-er 6707  df-map 6824  df-en 6915  df-dom 6916  df-fin 6917  df-pnf 8221  df-mnf 8222  df-xr 8223  df-ltxr 8224  df-le 8225  df-sub 8357  df-neg 8358  df-inn 9149  df-2 9207  df-3 9208  df-4 9209  df-5 9210  df-6 9211  df-7 9212  df-8 9213  df-9 9214  df-n0 9408  df-z 9485  df-dec 9617  df-uz 9761  df-fz 10249  df-fzo 10383  df-ihash 11044  df-word 11123  df-substr 11236  df-pfx 11263  df-ndx 13108  df-slot 13109  df-base 13111  df-edgf 15885  df-vtx 15894  df-iedg 15895  df-wlks 16198

This theorem is referenced by:  trlres  16270  eupthres  16337