Theorem wwlksnextsur 26795

Description: Lemma for wwlksnextbij 26797. (Contributed by Alexander van der Vekens, 7-Aug-2018.) (Revised by AV, 18-Apr-2021.)

Hypotheses

Ref	Expression
wwlksnextbij0.v	|- V = (Vtx ` G )
wwlksnextbij0.e	|- E = (Edg ` G )
wwlksnextbij0.d	|- D = { w e. Word V | ( ( # ` w ) = ( N + 2 ) /\ ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) }
wwlksnextbij.r	|- R = { n e. V | { ( lastS ` W ) , n } e. E }
wwlksnextbij.f	|- F = ( t e. D |-> ( lastS ` t ) )

Assertion

Ref	Expression
wwlksnextsur	|- ( W e. ( N WWalksN G ) -> F : D -onto-> R )

Distinct variable groups:   w, G   w, N   w, W   t, D   n, E, w   t, N, w   t, R   n, V, w   n, W   t, n, N, w

Allowed substitution hints:   D( w, n)   R( w, n)   E( t)   F( w, t, n)   G( t, n)   V( t)   W( t)

Proof of Theorem wwlksnextsur

Dummy variables i d r are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		wwlksnextbij0.v	. . . 4 |- V = (Vtx ` G )
2	1	wwlknbp 26733	. . 3 |- ( W e. ( N WWalksN G ) -> ( G e. _V /\ N e. NN0 /\ W e. Word V ) )
3		simp2 1062	. . 3 |- ( ( G e. _V /\ N e. NN0 /\ W e. Word V ) -> N e. NN0 )
4		wwlksnextbij0.e	. . . 4 |- E = (Edg ` G )
5		wwlksnextbij0.d	. . . 4 |- D = { w e. Word V | ( ( # ` w ) = ( N + 2 ) /\ ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) }
6		wwlksnextbij.r	. . . 4 |- R = { n e. V | { ( lastS ` W ) , n } e. E }
7		wwlksnextbij.f	. . . 4 |- F = ( t e. D |-> ( lastS ` t ) )
8	1, 4, 5, 6, 7	wwlksnextfun 26793	. . 3 |- ( N e. NN0 -> F : D --> R )
9	2, 3, 8	3syl 18	. 2 |- ( W e. ( N WWalksN G ) -> F : D --> R )
10		preq2 4269	. . . . . 6 |- ( n = r -> { ( lastS ` W ) , n } = { ( lastS ` W ) , r } )
11	10	eleq1d 2686	. . . . 5 |- ( n = r -> ( { ( lastS ` W ) , n } e. E <-> { ( lastS ` W ) , r } e. E ) )
12	11, 6	elrab2 3366	. . . 4 |- ( r e. R <-> ( r e. V /\ { ( lastS ` W ) , r } e. E ) )
13	1, 4	wwlksnext 26788	. . . . . . . . . . 11 |- ( ( W e. ( N WWalksN G ) /\ r e. V /\ { ( lastS ` W ) , r } e. E ) -> ( W ++ <" r "> ) e. ( ( N + 1 ) WWalksN G ) )
14	13	3expb 1266	. . . . . . . . . 10 |- ( ( W e. ( N WWalksN G ) /\ ( r e. V /\ { ( lastS ` W ) , r } e. E ) ) -> ( W ++ <" r "> ) e. ( ( N + 1 ) WWalksN G ) )
15		s1cl 13382	. . . . . . . . . . . . . . . . . 18 |- ( r e. V -> <" r "> e. Word V )
16		swrdccat1 13457	. . . . . . . . . . . . . . . . . 18 |- ( ( W e. Word V /\ <" r "> e. Word V ) -> ( ( W ++ <" r "> ) substr <. 0 , ( # ` W ) >. ) = W )
17	15, 16	sylan2 491	. . . . . . . . . . . . . . . . 17 |- ( ( W e. Word V /\ r e. V ) -> ( ( W ++ <" r "> ) substr <. 0 , ( # ` W ) >. ) = W )
18	17	ex 450	. . . . . . . . . . . . . . . 16 |- ( W e. Word V -> ( r e. V -> ( ( W ++ <" r "> ) substr <. 0 , ( # ` W ) >. ) = W ) )
19	18	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( W e. Word V /\ ( # ` W ) = ( N + 1 ) ) -> ( r e. V -> ( ( W ++ <" r "> ) substr <. 0 , ( # ` W ) >. ) = W ) )
20		opeq2 4403	. . . . . . . . . . . . . . . . . . 19 |- ( ( N + 1 ) = ( # ` W ) -> <. 0 , ( N + 1 ) >. = <. 0 , ( # ` W ) >. )
21	20	eqcoms 2630	. . . . . . . . . . . . . . . . . 18 |- ( ( # ` W ) = ( N + 1 ) -> <. 0 , ( N + 1 ) >. = <. 0 , ( # ` W ) >. )
22	21	oveq2d 6666	. . . . . . . . . . . . . . . . 17 |- ( ( # ` W ) = ( N + 1 ) -> ( ( W ++ <" r "> ) substr <. 0 , ( N + 1 ) >. ) = ( ( W ++ <" r "> ) substr <. 0 , ( # ` W ) >. ) )
23	22	eqeq1d 2624	. . . . . . . . . . . . . . . 16 |- ( ( # ` W ) = ( N + 1 ) -> ( ( ( W ++ <" r "> ) substr <. 0 , ( N + 1 ) >. ) = W <-> ( ( W ++ <" r "> ) substr <. 0 , ( # ` W ) >. ) = W ) )
24	23	adantl 482	. . . . . . . . . . . . . . 15 |- ( ( W e. Word V /\ ( # ` W ) = ( N + 1 ) ) -> ( ( ( W ++ <" r "> ) substr <. 0 , ( N + 1 ) >. ) = W <-> ( ( W ++ <" r "> ) substr <. 0 , ( # ` W ) >. ) = W ) )
25	19, 24	sylibrd 249	. . . . . . . . . . . . . 14 |- ( ( W e. Word V /\ ( # ` W ) = ( N + 1 ) ) -> ( r e. V -> ( ( W ++ <" r "> ) substr <. 0 , ( N + 1 ) >. ) = W ) )
26	25	3adant3 1081	. . . . . . . . . . . . 13 |- ( ( W e. Word V /\ ( # ` W ) = ( N + 1 ) /\ A. i e. ( 0..^ N ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E ) -> ( r e. V -> ( ( W ++ <" r "> ) substr <. 0 , ( N + 1 ) >. ) = W ) )
27	1, 4	wwlknp 26734	. . . . . . . . . . . . 13 |- ( W e. ( N WWalksN G ) -> ( W e. Word V /\ ( # ` W ) = ( N + 1 ) /\ A. i e. ( 0..^ N ) { ( W ` i ) , ( W ` ( i + 1 ) ) } e. E ) )
28	26, 27	syl11 33	. . . . . . . . . . . 12 |- ( r e. V -> ( W e. ( N WWalksN G ) -> ( ( W ++ <" r "> ) substr <. 0 , ( N + 1 ) >. ) = W ) )
29	28	adantr 481	. . . . . . . . . . 11 |- ( ( r e. V /\ { ( lastS ` W ) , r } e. E ) -> ( W e. ( N WWalksN G ) -> ( ( W ++ <" r "> ) substr <. 0 , ( N + 1 ) >. ) = W ) )
30	29	impcom 446	. . . . . . . . . 10 |- ( ( W e. ( N WWalksN G ) /\ ( r e. V /\ { ( lastS ` W ) , r } e. E ) ) -> ( ( W ++ <" r "> ) substr <. 0 , ( N + 1 ) >. ) = W )
31		lswccats1 13411	. . . . . . . . . . . . . . . . . . 19 |- ( ( W e. Word V /\ r e. V ) -> ( lastS ` ( W ++ <" r "> ) ) = r )
32	31	eqcomd 2628	. . . . . . . . . . . . . . . . . 18 |- ( ( W e. Word V /\ r e. V ) -> r = ( lastS ` ( W ++ <" r "> ) ) )
33	32	ex 450	. . . . . . . . . . . . . . . . 17 |- ( W e. Word V -> ( r e. V -> r = ( lastS ` ( W ++ <" r "> ) ) ) )
34	33	3ad2ant3 1084	. . . . . . . . . . . . . . . 16 |- ( ( G e. _V /\ N e. NN0 /\ W e. Word V ) -> ( r e. V -> r = ( lastS ` ( W ++ <" r "> ) ) ) )
35	2, 34	syl 17	. . . . . . . . . . . . . . 15 |- ( W e. ( N WWalksN G ) -> ( r e. V -> r = ( lastS ` ( W ++ <" r "> ) ) ) )
36	35	imp 445	. . . . . . . . . . . . . 14 |- ( ( W e. ( N WWalksN G ) /\ r e. V ) -> r = ( lastS ` ( W ++ <" r "> ) ) )
37	36	preq2d 4275	. . . . . . . . . . . . 13 |- ( ( W e. ( N WWalksN G ) /\ r e. V ) -> { ( lastS ` W ) , r } = { ( lastS ` W ) , ( lastS ` ( W ++ <" r "> ) ) } )
38	37	eleq1d 2686	. . . . . . . . . . . 12 |- ( ( W e. ( N WWalksN G ) /\ r e. V ) -> ( { ( lastS ` W ) , r } e. E <-> { ( lastS ` W ) , ( lastS ` ( W ++ <" r "> ) ) } e. E ) )
39	38	biimpd 219	. . . . . . . . . . 11 |- ( ( W e. ( N WWalksN G ) /\ r e. V ) -> ( { ( lastS ` W ) , r } e. E -> { ( lastS ` W ) , ( lastS ` ( W ++ <" r "> ) ) } e. E ) )
40	39	impr 649	. . . . . . . . . 10 |- ( ( W e. ( N WWalksN G ) /\ ( r e. V /\ { ( lastS ` W ) , r } e. E ) ) -> { ( lastS ` W ) , ( lastS ` ( W ++ <" r "> ) ) } e. E )
41	14, 30, 40	jca32 558	. . . . . . . . 9 |- ( ( W e. ( N WWalksN G ) /\ ( r e. V /\ { ( lastS ` W ) , r } e. E ) ) -> ( ( W ++ <" r "> ) e. ( ( N + 1 ) WWalksN G ) /\ ( ( ( W ++ <" r "> ) substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` ( W ++ <" r "> ) ) } e. E ) ) )
42	34, 2	syl11 33	. . . . . . . . . . 11 |- ( r e. V -> ( W e. ( N WWalksN G ) -> r = ( lastS ` ( W ++ <" r "> ) ) ) )
43	42	adantr 481	. . . . . . . . . 10 |- ( ( r e. V /\ { ( lastS ` W ) , r } e. E ) -> ( W e. ( N WWalksN G ) -> r = ( lastS ` ( W ++ <" r "> ) ) ) )
44	43	impcom 446	. . . . . . . . 9 |- ( ( W e. ( N WWalksN G ) /\ ( r e. V /\ { ( lastS ` W ) , r } e. E ) ) -> r = ( lastS ` ( W ++ <" r "> ) ) )
45		ovexd 6680	. . . . . . . . . 10 |- ( ( W e. ( N WWalksN G ) /\ ( r e. V /\ { ( lastS ` W ) , r } e. E ) ) -> ( W ++ <" r "> ) e. _V )
46		eleq1 2689	. . . . . . . . . . . . . . 15 |- ( d = ( W ++ <" r "> ) -> ( d e. ( ( N + 1 ) WWalksN G ) <-> ( W ++ <" r "> ) e. ( ( N + 1 ) WWalksN G ) ) )
47		oveq1 6657	. . . . . . . . . . . . . . . . 17 |- ( d = ( W ++ <" r "> ) -> ( d substr <. 0 , ( N + 1 ) >. ) = ( ( W ++ <" r "> ) substr <. 0 , ( N + 1 ) >. ) )
48	47	eqeq1d 2624	. . . . . . . . . . . . . . . 16 |- ( d = ( W ++ <" r "> ) -> ( ( d substr <. 0 , ( N + 1 ) >. ) = W <-> ( ( W ++ <" r "> ) substr <. 0 , ( N + 1 ) >. ) = W ) )
49		fveq2 6191	. . . . . . . . . . . . . . . . . 18 |- ( d = ( W ++ <" r "> ) -> ( lastS ` d ) = ( lastS ` ( W ++ <" r "> ) ) )
50	49	preq2d 4275	. . . . . . . . . . . . . . . . 17 |- ( d = ( W ++ <" r "> ) -> { ( lastS ` W ) , ( lastS ` d ) } = { ( lastS ` W ) , ( lastS ` ( W ++ <" r "> ) ) } )
51	50	eleq1d 2686	. . . . . . . . . . . . . . . 16 |- ( d = ( W ++ <" r "> ) -> ( { ( lastS ` W ) , ( lastS ` d ) } e. E <-> { ( lastS ` W ) , ( lastS ` ( W ++ <" r "> ) ) } e. E ) )
52	48, 51	anbi12d 747	. . . . . . . . . . . . . . 15 |- ( d = ( W ++ <" r "> ) -> ( ( ( d substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` d ) } e. E ) <-> ( ( ( W ++ <" r "> ) substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` ( W ++ <" r "> ) ) } e. E ) ) )
53	46, 52	anbi12d 747	. . . . . . . . . . . . . 14 |- ( d = ( W ++ <" r "> ) -> ( ( d e. ( ( N + 1 ) WWalksN G ) /\ ( ( d substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` d ) } e. E ) ) <-> ( ( W ++ <" r "> ) e. ( ( N + 1 ) WWalksN G ) /\ ( ( ( W ++ <" r "> ) substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` ( W ++ <" r "> ) ) } e. E ) ) ) )
54	49	eqeq2d 2632	. . . . . . . . . . . . . 14 |- ( d = ( W ++ <" r "> ) -> ( r = ( lastS ` d ) <-> r = ( lastS ` ( W ++ <" r "> ) ) ) )
55	53, 54	anbi12d 747	. . . . . . . . . . . . 13 |- ( d = ( W ++ <" r "> ) -> ( ( ( d e. ( ( N + 1 ) WWalksN G ) /\ ( ( d substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` d ) } e. E ) ) /\ r = ( lastS ` d ) ) <-> ( ( ( W ++ <" r "> ) e. ( ( N + 1 ) WWalksN G ) /\ ( ( ( W ++ <" r "> ) substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` ( W ++ <" r "> ) ) } e. E ) ) /\ r = ( lastS ` ( W ++ <" r "> ) ) ) ) )
56	55	bicomd 213	. . . . . . . . . . . 12 |- ( d = ( W ++ <" r "> ) -> ( ( ( ( W ++ <" r "> ) e. ( ( N + 1 ) WWalksN G ) /\ ( ( ( W ++ <" r "> ) substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` ( W ++ <" r "> ) ) } e. E ) ) /\ r = ( lastS ` ( W ++ <" r "> ) ) ) <-> ( ( d e. ( ( N + 1 ) WWalksN G ) /\ ( ( d substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` d ) } e. E ) ) /\ r = ( lastS ` d ) ) ) )
57	56	adantl 482	. . . . . . . . . . 11 |- ( ( ( W e. ( N WWalksN G ) /\ ( r e. V /\ { ( lastS ` W ) , r } e. E ) ) /\ d = ( W ++ <" r "> ) ) -> ( ( ( ( W ++ <" r "> ) e. ( ( N + 1 ) WWalksN G ) /\ ( ( ( W ++ <" r "> ) substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` ( W ++ <" r "> ) ) } e. E ) ) /\ r = ( lastS ` ( W ++ <" r "> ) ) ) <-> ( ( d e. ( ( N + 1 ) WWalksN G ) /\ ( ( d substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` d ) } e. E ) ) /\ r = ( lastS ` d ) ) ) )
58	57	biimpd 219	. . . . . . . . . 10 |- ( ( ( W e. ( N WWalksN G ) /\ ( r e. V /\ { ( lastS ` W ) , r } e. E ) ) /\ d = ( W ++ <" r "> ) ) -> ( ( ( ( W ++ <" r "> ) e. ( ( N + 1 ) WWalksN G ) /\ ( ( ( W ++ <" r "> ) substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` ( W ++ <" r "> ) ) } e. E ) ) /\ r = ( lastS ` ( W ++ <" r "> ) ) ) -> ( ( d e. ( ( N + 1 ) WWalksN G ) /\ ( ( d substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` d ) } e. E ) ) /\ r = ( lastS ` d ) ) ) )
59	45, 58	spcimedv 3292	. . . . . . . . 9 |- ( ( W e. ( N WWalksN G ) /\ ( r e. V /\ { ( lastS ` W ) , r } e. E ) ) -> ( ( ( ( W ++ <" r "> ) e. ( ( N + 1 ) WWalksN G ) /\ ( ( ( W ++ <" r "> ) substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` ( W ++ <" r "> ) ) } e. E ) ) /\ r = ( lastS ` ( W ++ <" r "> ) ) ) -> E. d ( ( d e. ( ( N + 1 ) WWalksN G ) /\ ( ( d substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` d ) } e. E ) ) /\ r = ( lastS ` d ) ) ) )
60	41, 44, 59	mp2and 715	. . . . . . . 8 |- ( ( W e. ( N WWalksN G ) /\ ( r e. V /\ { ( lastS ` W ) , r } e. E ) ) -> E. d ( ( d e. ( ( N + 1 ) WWalksN G ) /\ ( ( d substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` d ) } e. E ) ) /\ r = ( lastS ` d ) ) )
61		oveq1 6657	. . . . . . . . . . . . 13 |- ( w = d -> ( w substr <. 0 , ( N + 1 ) >. ) = ( d substr <. 0 , ( N + 1 ) >. ) )
62	61	eqeq1d 2624	. . . . . . . . . . . 12 |- ( w = d -> ( ( w substr <. 0 , ( N + 1 ) >. ) = W <-> ( d substr <. 0 , ( N + 1 ) >. ) = W ) )
63		fveq2 6191	. . . . . . . . . . . . . 14 |- ( w = d -> ( lastS ` w ) = ( lastS ` d ) )
64	63	preq2d 4275	. . . . . . . . . . . . 13 |- ( w = d -> { ( lastS ` W ) , ( lastS ` w ) } = { ( lastS ` W ) , ( lastS ` d ) } )
65	64	eleq1d 2686	. . . . . . . . . . . 12 |- ( w = d -> ( { ( lastS ` W ) , ( lastS ` w ) } e. E <-> { ( lastS ` W ) , ( lastS ` d ) } e. E ) )
66	62, 65	anbi12d 747	. . . . . . . . . . 11 |- ( w = d -> ( ( ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) <-> ( ( d substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` d ) } e. E ) ) )
67	66	elrab 3363	. . . . . . . . . 10 |- ( d e. { w e. ( ( N + 1 ) WWalksN G ) | ( ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) } <-> ( d e. ( ( N + 1 ) WWalksN G ) /\ ( ( d substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` d ) } e. E ) ) )
68	67	anbi1i 731	. . . . . . . . 9 |- ( ( d e. { w e. ( ( N + 1 ) WWalksN G ) | ( ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) } /\ r = ( lastS ` d ) ) <-> ( ( d e. ( ( N + 1 ) WWalksN G ) /\ ( ( d substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` d ) } e. E ) ) /\ r = ( lastS ` d ) ) )
69	68	exbii 1774	. . . . . . . 8 |- ( E. d ( d e. { w e. ( ( N + 1 ) WWalksN G ) | ( ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) } /\ r = ( lastS ` d ) ) <-> E. d ( ( d e. ( ( N + 1 ) WWalksN G ) /\ ( ( d substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` d ) } e. E ) ) /\ r = ( lastS ` d ) ) )
70	60, 69	sylibr 224	. . . . . . 7 |- ( ( W e. ( N WWalksN G ) /\ ( r e. V /\ { ( lastS ` W ) , r } e. E ) ) -> E. d ( d e. { w e. ( ( N + 1 ) WWalksN G ) | ( ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) } /\ r = ( lastS ` d ) ) )
71		df-rex 2918	. . . . . . 7 |- ( E. d e. { w e. ( ( N + 1 ) WWalksN G ) | ( ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) } r = ( lastS ` d ) <-> E. d ( d e. { w e. ( ( N + 1 ) WWalksN G ) | ( ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) } /\ r = ( lastS ` d ) ) )
72	70, 71	sylibr 224	. . . . . 6 |- ( ( W e. ( N WWalksN G ) /\ ( r e. V /\ { ( lastS ` W ) , r } e. E ) ) -> E. d e. { w e. ( ( N + 1 ) WWalksN G ) | ( ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) } r = ( lastS ` d ) )
73	1, 4, 5	wwlksnextwrd 26792	. . . . . . . 8 |- ( W e. ( N WWalksN G ) -> D = { w e. ( ( N + 1 ) WWalksN G ) | ( ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) } )
74	73	adantr 481	. . . . . . 7 |- ( ( W e. ( N WWalksN G ) /\ ( r e. V /\ { ( lastS ` W ) , r } e. E ) ) -> D = { w e. ( ( N + 1 ) WWalksN G ) | ( ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) } )
75	74	rexeqdv 3145	. . . . . 6 |- ( ( W e. ( N WWalksN G ) /\ ( r e. V /\ { ( lastS ` W ) , r } e. E ) ) -> ( E. d e. D r = ( lastS ` d ) <-> E. d e. { w e. ( ( N + 1 ) WWalksN G ) | ( ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) } r = ( lastS ` d ) ) )
76	72, 75	mpbird 247	. . . . 5 |- ( ( W e. ( N WWalksN G ) /\ ( r e. V /\ { ( lastS ` W ) , r } e. E ) ) -> E. d e. D r = ( lastS ` d ) )
77		fveq2 6191	. . . . . . . 8 |- ( t = d -> ( lastS ` t ) = ( lastS ` d ) )
78		fvex 6201	. . . . . . . 8 |- ( lastS ` d ) e. _V
79	77, 7, 78	fvmpt 6282	. . . . . . 7 |- ( d e. D -> ( F ` d ) = ( lastS ` d ) )
80	79	eqeq2d 2632	. . . . . 6 |- ( d e. D -> ( r = ( F ` d ) <-> r = ( lastS ` d ) ) )
81	80	rexbiia 3040	. . . . 5 |- ( E. d e. D r = ( F ` d ) <-> E. d e. D r = ( lastS ` d ) )
82	76, 81	sylibr 224	. . . 4 |- ( ( W e. ( N WWalksN G ) /\ ( r e. V /\ { ( lastS ` W ) , r } e. E ) ) -> E. d e. D r = ( F ` d ) )
83	12, 82	sylan2b 492	. . 3 |- ( ( W e. ( N WWalksN G ) /\ r e. R ) -> E. d e. D r = ( F ` d ) )
84	83	ralrimiva 2966	. 2 |- ( W e. ( N WWalksN G ) -> A. r e. R E. d e. D r = ( F ` d ) )
85		dffo3 6374	. 2 |- ( F : D -onto-> R <-> ( F : D --> R /\ A. r e. R E. d e. D r = ( F ` d ) ) )
86	9, 84, 85	sylanbrc 698	1 |- ( W e. ( N WWalksN G ) -> F : D -onto-> R )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   {cpr 4179   <.cop 4183   |-> cmpt 4729   -->wf 5884   -onto->wfo 5886   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   2c2 11070   NN0cn0 11292  ..^cfzo 12465   #chash 13117  Word cword 13291   lastS clsw 13292   ++ cconcat 13293   <"cs1 13294   substr csubstr 13295  Vtxcvtx 25874  Edgcedg 25939   WWalksN cwwlksn 26718

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-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-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-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-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-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-lsw 13300  df-concat 13301  df-s1 13302  df-substr 13303  df-wwlks 26722  df-wwlksn 26723

This theorem is referenced by:  wwlksnextbij0  26796