Theorem wlkres 16165

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 16118	. . . . 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 10413	. . . . 5 |- ( N e. ( 0..^ (♯ ` F ) ) -> N e. NN0 )
7	5, 6	syl 14	. . . 4 |- ( ph -> N e. NN0 )
8		pfxwrdsymbg 11258	. . . 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 4929	. . . . 5 |- ( ph -> dom (iEdg ` S ) = dom ( I |` ( F " ( 0..^ N ) ) ) )
14		wrdf 11106	. . . . . . 7 |- ( F e. Word dom I -> F : ( 0..^ (♯ ` F ) ) --> dom I )
15		fimass 5493	. . . . . . 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 5031	. . . . . 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 2262	. . . 4 |- ( ph -> dom (iEdg ` S ) = ( F " ( 0..^ N ) ) )
20		wrdeq 11122	. . . 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 2313	. 2 |- ( ph -> H e. Word dom (iEdg ` S ) )
23		wlkres.v	. . . . . . . 8 |- V = (Vtx ` G )
24	23	wlkp 16122	. . . . . . 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 5466	. . . . . 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 10389	. . . . . . 7 |- ( 0..^ (♯ ` F ) ) C_ ( 0 ... (♯ ` F ) )
30	29, 5	sselid 3223	. . . . . 6 |- ( ph -> N e. ( 0 ... (♯ ` F ) ) )
31		elfzuz3 10245	. . . . . 6 |- ( N e. ( 0 ... (♯ ` F ) ) -> (♯ ` F ) e. ( ZZ>= ` N ) )
32		fzss2 10287	. . . . . 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 5508	. . . 4 |- ( ph -> ( P |` ( 0 ... N ) ) : ( 0 ... N ) --> (Vtx ` S ) )
35	10	fveq2i 5636	. . . . . . 7 |- (♯ ` H ) = (♯ ` ( F prefix N ) )
36		pfxlen 11253	. . . . . . . 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 2274	. . . . . 6 |- ( ph -> (♯ ` H ) = N )
39	38	oveq2d 6027	. . . . 5 |- ( ph -> ( 0 ... (♯ ` H ) ) = ( 0 ... N ) )
40	39	feq2d 5465	. . . 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 5470	. . 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 16115	. . . . . 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 6027	. . . . . . . . . . 11 |- ( ph -> ( 0..^ (♯ ` H ) ) = ( 0..^ N ) )
49	48	eleq2d 2299	. . . . . . . . . 10 |- ( ph -> ( x e. ( 0..^ (♯ ` H ) ) <-> x e. ( 0..^ N ) ) )
50	42	fveq1i 5634	. . . . . . . . . . . . 13 |- ( Q ` x ) = ( ( P |` ( 0 ... N ) ) ` x )
51		fzossfz 10389	. . . . . . . . . . . . . . . 16 |- ( 0..^ N ) C_ ( 0 ... N )
52	51	a1i 9	. . . . . . . . . . . . . . 15 |- ( ph -> ( 0..^ N ) C_ ( 0 ... N ) )
53	52	sselda 3225	. . . . . . . . . . . . . 14 |- ( ( ph /\ x e. ( 0..^ N ) ) -> x e. ( 0 ... N ) )
54	53	fvresd 5658	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. ( 0..^ N ) ) -> ( ( P |` ( 0 ... N ) ) ` x ) = ( P ` x ) )
55	50, 54	eqtr2id 2275	. . . . . . . . . . . 12 |- ( ( ph /\ x e. ( 0..^ N ) ) -> ( P ` x ) = ( Q ` x ) )
56	42	fveq1i 5634	. . . . . . . . . . . . 13 |- ( Q ` ( x + 1 ) ) = ( ( P |` ( 0 ... N ) ) ` ( x + 1 ) )
57		fzofzp1 10460	. . . . . . . . . . . . . . 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 5658	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. ( 0..^ N ) ) -> ( ( P |` ( 0 ... N ) ) ` ( x + 1 ) ) = ( P ` ( x + 1 ) ) )
60	56, 59	eqtr2id 2275	. . . . . . . . . . . 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 5481	. . . . . . . . . . . . . . . . 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 5483	. . . . . . . . . . . . . . . . . 18 |- ( F : ( 0..^ (♯ ` F ) ) --> dom I -> dom F = ( 0..^ (♯ ` F ) ) )
70		elfzouz2 10385	. . . . . . . . . . . . . . . . . . . 20 |- ( N e. ( 0..^ (♯ ` F ) ) -> (♯ ` F ) e. ( ZZ>= ` N ) )
71		fzoss2 10397	. . . . . . . . . . . . . . . . . . . 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 3249	. . . . . . . . . . . . . . . . . . 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 5884	. . . . . . . . . . . . . 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 2235	. . . . . . . . . . . 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 5634	. . . . . . . . . . 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 11249	. . . . . . . . . . . . 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 5635	. . . . . . . . . . 11 |- ( ( ph /\ x e. ( 0..^ (♯ ` H ) ) ) -> ( ( F prefix N ) ` x ) = ( ( F |` ( 0..^ N ) ) ` x ) )
92	86, 91	eqtrid 2274	. . . . . . . . . 10 |- ( ( ph /\ x e. ( 0..^ (♯ ` H ) ) ) -> ( H ` x ) = ( ( F |` ( 0..^ N ) ) ` x ) )
93	85, 92	fveq12d 5640	. . . . . . . . 9 |- ( ( ph /\ x e. ( 0..^ (♯ ` H ) ) ) -> ( (iEdg ` S ) ` ( H ` x ) ) = ( ( I |` ( F " ( 0..^ N ) ) ) ` ( ( F |` ( 0..^ N ) ) ` x ) ) )
94	84, 93	eqtr4d 2265	. . . . . . . 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 5637	. . . . . . . . . . 11 |- ( ph -> ( ZZ>= ` (♯ ` H ) ) = ( ZZ>= ` N ) )
98	96, 97	eleqtrrd 2309	. . . . . . . . . 10 |- ( ph -> (♯ ` F ) e. ( ZZ>= ` (♯ ` H ) ) )
99		fzoss2 10397	. . . . . . . . . 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 3225	. . . . . . . 8 |- ( ( ph /\ x e. ( 0..^ (♯ ` H ) ) ) -> x e. ( 0..^ (♯ ` F ) ) )
102		wkslem1 16108	. . . . . . . . 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 2904	. . . . . . . 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 2242	. . . . . . . . . 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 3678	. . . . . . . . . . . 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 2244	. . . . . . . . 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 3746	. . . . . . . . . . 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 3256	. . . . . . . . 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 998	. . . . . . . 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 1044	. . . 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 2603	. 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 16164	. . 3 |- ( ph -> S e. _V )
123		eqid 2229	. . . 4 |- (Vtx ` S ) = (Vtx ` S )
124		eqid 2229	. . . 4 |- (iEdg ` S ) = (iEdg ` S )
125	123, 124	iswlkg 16117	. . 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 1204	1 |- ( ph -> H (Walks ` S ) Q )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105  if-wif 983   /\ w3a 1002   = wceq 1395   e. wcel 2200   A.wral 2508   _Vcvv 2800   C_ wss 3198   {csn 3667   {cpr 3668   class class class wbr 4084   dom cdm 4721   |` cres 4723   "cima 4724   Fun wfun 5316   -->wf 5318   ` cfv 5322  (class class class)co 6011   0cc0 8020   1c1 8021   + caddc 8023   NN0cn0 9390   ZZ>=cuz 9743   ...cfz 10231  ..^cfzo 10365  ♯chash 11025  Word cword 11100   prefix cpfx 11240  Vtxcvtx 15850  iEdgciedg 15851  Walkscwlks 16105

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4200  ax-sep 4203  ax-nul 4211  ax-pow 4260  ax-pr 4295  ax-un 4526  ax-setind 4631  ax-iinf 4682  ax-cnex 8111  ax-resscn 8112  ax-1cn 8113  ax-1re 8114  ax-icn 8115  ax-addcl 8116  ax-addrcl 8117  ax-mulcl 8118  ax-addcom 8120  ax-mulcom 8121  ax-addass 8122  ax-mulass 8123  ax-distr 8124  ax-i2m1 8125  ax-0lt1 8126  ax-1rid 8127  ax-0id 8128  ax-rnegex 8129  ax-cnre 8131  ax-pre-ltirr 8132  ax-pre-ltwlin 8133  ax-pre-lttrn 8134  ax-pre-apti 8135  ax-pre-ltadd 8136

This theorem depends on definitions:  df-bi 117  df-dc 840  df-ifp 984  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3890  df-int 3925  df-iun 3968  df-br 4085  df-opab 4147  df-mpt 4148  df-tr 4184  df-id 4386  df-iord 4459  df-on 4461  df-ilim 4462  df-suc 4464  df-iom 4685  df-xp 4727  df-rel 4728  df-cnv 4729  df-co 4730  df-dm 4731  df-rn 4732  df-res 4733  df-ima 4734  df-iota 5282  df-fun 5324  df-fn 5325  df-f 5326  df-f1 5327  df-fo 5328  df-f1o 5329  df-fv 5330  df-riota 5964  df-ov 6014  df-oprab 6015  df-mpo 6016  df-1st 6296  df-2nd 6297  df-recs 6464  df-frec 6550  df-1o 6575  df-er 6695  df-map 6812  df-en 6903  df-dom 6904  df-fin 6905  df-pnf 8204  df-mnf 8205  df-xr 8206  df-ltxr 8207  df-le 8208  df-sub 8340  df-neg 8341  df-inn 9132  df-2 9190  df-3 9191  df-4 9192  df-5 9193  df-6 9194  df-7 9195  df-8 9196  df-9 9197  df-n0 9391  df-z 9468  df-dec 9600  df-uz 9744  df-fz 10232  df-fzo 10366  df-ihash 11026  df-word 11101  df-substr 11214  df-pfx 11241  df-ndx 13072  df-slot 13073  df-base 13075  df-edgf 15843  df-vtx 15852  df-iedg 15853  df-wlks 16106

This theorem is referenced by:  trlres  16175