Theorem elwwlks2 26861

Description: A walk of length 2 between two vertices as length 3 string in a pseudograph. (Contributed by Alexander van der Vekens, 21-Feb-2018.) (Revised by AV, 17-May-2021.)

Hypothesis

Ref	Expression
elwwlks2.v	|- V = (Vtx ` G )

Assertion

Ref	Expression
elwwlks2	|- ( G e. UPGraph -> ( W e. ( 2 WWalksN G ) <-> E. a e. V E. b e. V E. c e. V ( W = <" a b c "> /\ E. f E. p ( f (Walks ` G ) p /\ ( # ` f ) = 2 /\ ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) ) ) ) )

Distinct variable groups:   G, a, b, c, f, p   V, a, b, c, f, p   W, a, b, c, f, p

Proof of Theorem elwwlks2

Step	Hyp	Ref	Expression
1		2nn0 11309	. . 3 |- 2 e. NN0
2		elwwlks2.v	. . . 4 |- V = (Vtx ` G )
3	2	wwlksnwwlksnon 26810	. . 3 |- ( ( 2 e. NN0 /\ G e. UPGraph ) -> ( W e. ( 2 WWalksN G ) <-> E. a e. V E. c e. V W e. ( a ( 2 WWalksNOn G ) c ) ) )
4	1, 3	mpan 706	. 2 |- ( G e. UPGraph -> ( W e. ( 2 WWalksN G ) <-> E. a e. V E. c e. V W e. ( a ( 2 WWalksNOn G ) c ) ) )
5	2	elwwlks2on 26852	. . . 4 |- ( ( G e. UPGraph /\ a e. V /\ c e. V ) -> ( W e. ( a ( 2 WWalksNOn G ) c ) <-> E. b e. V ( W = <" a b c "> /\ E. f ( f (Walks ` G ) W /\ ( # ` f ) = 2 ) ) ) )
6	5	3expb 1266	. . 3 |- ( ( G e. UPGraph /\ ( a e. V /\ c e. V ) ) -> ( W e. ( a ( 2 WWalksNOn G ) c ) <-> E. b e. V ( W = <" a b c "> /\ E. f ( f (Walks ` G ) W /\ ( # ` f ) = 2 ) ) ) )
7	6	2rexbidva 3056	. 2 |- ( G e. UPGraph -> ( E. a e. V E. c e. V W e. ( a ( 2 WWalksNOn G ) c ) <-> E. a e. V E. c e. V E. b e. V ( W = <" a b c "> /\ E. f ( f (Walks ` G ) W /\ ( # ` f ) = 2 ) ) ) )
8		rexcom 3099	. . . 4 |- ( E. c e. V E. b e. V ( W = <" a b c "> /\ E. f ( f (Walks ` G ) W /\ ( # ` f ) = 2 ) ) <-> E. b e. V E. c e. V ( W = <" a b c "> /\ E. f ( f (Walks ` G ) W /\ ( # ` f ) = 2 ) ) )
9		s3cli 13626	. . . . . . . . . 10 |- <" a b c "> e. Word _V
10	9	a1i 11	. . . . . . . . 9 |- ( ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ W = <" a b c "> ) -> <" a b c "> e. Word _V )
11		simplr 792	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ W = <" a b c "> ) /\ p = <" a b c "> ) -> W = <" a b c "> )
12		simpr 477	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ W = <" a b c "> ) /\ p = <" a b c "> ) -> p = <" a b c "> )
13	11, 12	eqtr4d 2659	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ W = <" a b c "> ) /\ p = <" a b c "> ) -> W = p )
14	13	breq2d 4665	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ W = <" a b c "> ) /\ p = <" a b c "> ) -> ( f (Walks ` G ) W <-> f (Walks ` G ) p ) )
15	14	biimpd 219	. . . . . . . . . . . . . 14 |- ( ( ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ W = <" a b c "> ) /\ p = <" a b c "> ) -> ( f (Walks ` G ) W -> f (Walks ` G ) p ) )
16	15	com12 32	. . . . . . . . . . . . 13 |- ( f (Walks ` G ) W -> ( ( ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ W = <" a b c "> ) /\ p = <" a b c "> ) -> f (Walks ` G ) p ) )
17	16	adantr 481	. . . . . . . . . . . 12 |- ( ( f (Walks ` G ) W /\ ( # ` f ) = 2 ) -> ( ( ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ W = <" a b c "> ) /\ p = <" a b c "> ) -> f (Walks ` G ) p ) )
18	17	impcom 446	. . . . . . . . . . 11 |- ( ( ( ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ W = <" a b c "> ) /\ p = <" a b c "> ) /\ ( f (Walks ` G ) W /\ ( # ` f ) = 2 ) ) -> f (Walks ` G ) p )
19		simprr 796	. . . . . . . . . . 11 |- ( ( ( ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ W = <" a b c "> ) /\ p = <" a b c "> ) /\ ( f (Walks ` G ) W /\ ( # ` f ) = 2 ) ) -> ( # ` f ) = 2 )
20		vex 3203	. . . . . . . . . . . . . . . 16 |- a e. _V
21		s3fv0 13636	. . . . . . . . . . . . . . . . 17 |- ( a e. _V -> ( <" a b c "> ` 0 ) = a )
22	21	eqcomd 2628	. . . . . . . . . . . . . . . 16 |- ( a e. _V -> a = ( <" a b c "> ` 0 ) )
23	20, 22	mp1i 13	. . . . . . . . . . . . . . 15 |- ( p = <" a b c "> -> a = ( <" a b c "> ` 0 ) )
24		fveq1 6190	. . . . . . . . . . . . . . 15 |- ( p = <" a b c "> -> ( p ` 0 ) = ( <" a b c "> ` 0 ) )
25	23, 24	eqtr4d 2659	. . . . . . . . . . . . . 14 |- ( p = <" a b c "> -> a = ( p ` 0 ) )
26		vex 3203	. . . . . . . . . . . . . . . 16 |- b e. _V
27		s3fv1 13637	. . . . . . . . . . . . . . . . 17 |- ( b e. _V -> ( <" a b c "> ` 1 ) = b )
28	27	eqcomd 2628	. . . . . . . . . . . . . . . 16 |- ( b e. _V -> b = ( <" a b c "> ` 1 ) )
29	26, 28	mp1i 13	. . . . . . . . . . . . . . 15 |- ( p = <" a b c "> -> b = ( <" a b c "> ` 1 ) )
30		fveq1 6190	. . . . . . . . . . . . . . 15 |- ( p = <" a b c "> -> ( p ` 1 ) = ( <" a b c "> ` 1 ) )
31	29, 30	eqtr4d 2659	. . . . . . . . . . . . . 14 |- ( p = <" a b c "> -> b = ( p ` 1 ) )
32		vex 3203	. . . . . . . . . . . . . . . 16 |- c e. _V
33		s3fv2 13638	. . . . . . . . . . . . . . . . 17 |- ( c e. _V -> ( <" a b c "> ` 2 ) = c )
34	33	eqcomd 2628	. . . . . . . . . . . . . . . 16 |- ( c e. _V -> c = ( <" a b c "> ` 2 ) )
35	32, 34	mp1i 13	. . . . . . . . . . . . . . 15 |- ( p = <" a b c "> -> c = ( <" a b c "> ` 2 ) )
36		fveq1 6190	. . . . . . . . . . . . . . 15 |- ( p = <" a b c "> -> ( p ` 2 ) = ( <" a b c "> ` 2 ) )
37	35, 36	eqtr4d 2659	. . . . . . . . . . . . . 14 |- ( p = <" a b c "> -> c = ( p ` 2 ) )
38	25, 31, 37	3jca 1242	. . . . . . . . . . . . 13 |- ( p = <" a b c "> -> ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) )
39	38	adantl 482	. . . . . . . . . . . 12 |- ( ( ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ W = <" a b c "> ) /\ p = <" a b c "> ) -> ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) )
40	39	adantr 481	. . . . . . . . . . 11 |- ( ( ( ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ W = <" a b c "> ) /\ p = <" a b c "> ) /\ ( f (Walks ` G ) W /\ ( # ` f ) = 2 ) ) -> ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) )
41	18, 19, 40	3jca 1242	. . . . . . . . . 10 |- ( ( ( ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ W = <" a b c "> ) /\ p = <" a b c "> ) /\ ( f (Walks ` G ) W /\ ( # ` f ) = 2 ) ) -> ( f (Walks ` G ) p /\ ( # ` f ) = 2 /\ ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) ) )
42	41	ex 450	. . . . . . . . 9 |- ( ( ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ W = <" a b c "> ) /\ p = <" a b c "> ) -> ( ( f (Walks ` G ) W /\ ( # ` f ) = 2 ) -> ( f (Walks ` G ) p /\ ( # ` f ) = 2 /\ ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) ) ) )
43	10, 42	spcimedv 3292	. . . . . . . 8 |- ( ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ W = <" a b c "> ) -> ( ( f (Walks ` G ) W /\ ( # ` f ) = 2 ) -> E. p ( f (Walks ` G ) p /\ ( # ` f ) = 2 /\ ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) ) ) )
44		wlklenvp1 26514	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( f (Walks ` G ) p -> ( # ` p ) = ( ( # ` f ) + 1 ) )
45		simpl 473	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( ( # ` p ) = ( ( # ` f ) + 1 ) /\ ( # ` f ) = 2 ) -> ( # ` p ) = ( ( # ` f ) + 1 ) )
46		oveq1 6657	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( ( # ` f ) = 2 -> ( ( # ` f ) + 1 ) = ( 2 + 1 ) )
47	46	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( ( # ` p ) = ( ( # ` f ) + 1 ) /\ ( # ` f ) = 2 ) -> ( ( # ` f ) + 1 ) = ( 2 + 1 ) )
48	45, 47	eqtrd 2656	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( ( # ` p ) = ( ( # ` f ) + 1 ) /\ ( # ` f ) = 2 ) -> ( # ` p ) = ( 2 + 1 ) )
49	48	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( f (Walks ` G ) p /\ ( ( # ` p ) = ( ( # ` f ) + 1 ) /\ ( # ` f ) = 2 ) ) -> ( # ` p ) = ( 2 + 1 ) )
50		2p1e3 11151	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( 2 + 1 ) = 3
51	49, 50	syl6eq 2672	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( f (Walks ` G ) p /\ ( ( # ` p ) = ( ( # ` f ) + 1 ) /\ ( # ` f ) = 2 ) ) -> ( # ` p ) = 3 )
52	51	exp32 631	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( f (Walks ` G ) p -> ( ( # ` p ) = ( ( # ` f ) + 1 ) -> ( ( # ` f ) = 2 -> ( # ` p ) = 3 ) ) )
53	44, 52	mpd 15	. . . . . . . . . . . . . . . . . . . . . 22 |- ( f (Walks ` G ) p -> ( ( # ` f ) = 2 -> ( # ` p ) = 3 ) )
54	53	adantr 481	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( f (Walks ` G ) p /\ ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ W = <" a b c "> ) ) -> ( ( # ` f ) = 2 -> ( # ` p ) = 3 ) )
55	54	imp 445	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( f (Walks ` G ) p /\ ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ W = <" a b c "> ) ) /\ ( # ` f ) = 2 ) -> ( # ` p ) = 3 )
56		eqcom 2629	. . . . . . . . . . . . . . . . . . . . . 22 |- ( a = ( p ` 0 ) <-> ( p ` 0 ) = a )
57	56	biimpi 206	. . . . . . . . . . . . . . . . . . . . 21 |- ( a = ( p ` 0 ) -> ( p ` 0 ) = a )
58		eqcom 2629	. . . . . . . . . . . . . . . . . . . . . 22 |- ( b = ( p ` 1 ) <-> ( p ` 1 ) = b )
59	58	biimpi 206	. . . . . . . . . . . . . . . . . . . . 21 |- ( b = ( p ` 1 ) -> ( p ` 1 ) = b )
60		eqcom 2629	. . . . . . . . . . . . . . . . . . . . . 22 |- ( c = ( p ` 2 ) <-> ( p ` 2 ) = c )
61	60	biimpi 206	. . . . . . . . . . . . . . . . . . . . 21 |- ( c = ( p ` 2 ) -> ( p ` 2 ) = c )
62	57, 59, 61	3anim123i 1247	. . . . . . . . . . . . . . . . . . . 20 |- ( ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) -> ( ( p ` 0 ) = a /\ ( p ` 1 ) = b /\ ( p ` 2 ) = c ) )
63	55, 62	anim12i 590	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( f (Walks ` G ) p /\ ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ W = <" a b c "> ) ) /\ ( # ` f ) = 2 ) /\ ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) ) -> ( ( # ` p ) = 3 /\ ( ( p ` 0 ) = a /\ ( p ` 1 ) = b /\ ( p ` 2 ) = c ) ) )
64	2	wlkpwrd 26513	. . . . . . . . . . . . . . . . . . . . . 22 |- ( f (Walks ` G ) p -> p e. Word V )
65		simpr 477	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( G e. UPGraph /\ a e. V ) -> a e. V )
66	65	anim1i 592	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) -> ( a e. V /\ ( b e. V /\ c e. V ) ) )
67		3anass 1042	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( a e. V /\ b e. V /\ c e. V ) <-> ( a e. V /\ ( b e. V /\ c e. V ) ) )
68	66, 67	sylibr 224	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) -> ( a e. V /\ b e. V /\ c e. V ) )
69	68	adantr 481	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ W = <" a b c "> ) -> ( a e. V /\ b e. V /\ c e. V ) )
70	64, 69	anim12i 590	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( f (Walks ` G ) p /\ ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ W = <" a b c "> ) ) -> ( p e. Word V /\ ( a e. V /\ b e. V /\ c e. V ) ) )
71	70	ad2antrr 762	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( f (Walks ` G ) p /\ ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ W = <" a b c "> ) ) /\ ( # ` f ) = 2 ) /\ ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) ) -> ( p e. Word V /\ ( a e. V /\ b e. V /\ c e. V ) ) )
72		eqwrds3 13704	. . . . . . . . . . . . . . . . . . . 20 |- ( ( p e. Word V /\ ( a e. V /\ b e. V /\ c e. V ) ) -> ( p = <" a b c "> <-> ( ( # ` p ) = 3 /\ ( ( p ` 0 ) = a /\ ( p ` 1 ) = b /\ ( p ` 2 ) = c ) ) ) )
73	71, 72	syl 17	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( f (Walks ` G ) p /\ ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ W = <" a b c "> ) ) /\ ( # ` f ) = 2 ) /\ ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) ) -> ( p = <" a b c "> <-> ( ( # ` p ) = 3 /\ ( ( p ` 0 ) = a /\ ( p ` 1 ) = b /\ ( p ` 2 ) = c ) ) ) )
74	63, 73	mpbird 247	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( f (Walks ` G ) p /\ ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ W = <" a b c "> ) ) /\ ( # ` f ) = 2 ) /\ ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) ) -> p = <" a b c "> )
75		simprr 796	. . . . . . . . . . . . . . . . . . 19 |- ( ( f (Walks ` G ) p /\ ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ W = <" a b c "> ) ) -> W = <" a b c "> )
76	75	ad2antrr 762	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( f (Walks ` G ) p /\ ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ W = <" a b c "> ) ) /\ ( # ` f ) = 2 ) /\ ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) ) -> W = <" a b c "> )
77	74, 76	eqtr4d 2659	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( f (Walks ` G ) p /\ ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ W = <" a b c "> ) ) /\ ( # ` f ) = 2 ) /\ ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) ) -> p = W )
78	77	breq2d 4665	. . . . . . . . . . . . . . . 16 |- ( ( ( ( f (Walks ` G ) p /\ ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ W = <" a b c "> ) ) /\ ( # ` f ) = 2 ) /\ ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) ) -> ( f (Walks ` G ) p <-> f (Walks ` G ) W ) )
79	78	biimpd 219	. . . . . . . . . . . . . . 15 |- ( ( ( ( f (Walks ` G ) p /\ ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ W = <" a b c "> ) ) /\ ( # ` f ) = 2 ) /\ ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) ) -> ( f (Walks ` G ) p -> f (Walks ` G ) W ) )
80		simplr 792	. . . . . . . . . . . . . . 15 |- ( ( ( ( f (Walks ` G ) p /\ ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ W = <" a b c "> ) ) /\ ( # ` f ) = 2 ) /\ ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) ) -> ( # ` f ) = 2 )
81	79, 80	jctird 567	. . . . . . . . . . . . . 14 |- ( ( ( ( f (Walks ` G ) p /\ ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ W = <" a b c "> ) ) /\ ( # ` f ) = 2 ) /\ ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) ) -> ( f (Walks ` G ) p -> ( f (Walks ` G ) W /\ ( # ` f ) = 2 ) ) )
82	81	exp41 638	. . . . . . . . . . . . 13 |- ( f (Walks ` G ) p -> ( ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ W = <" a b c "> ) -> ( ( # ` f ) = 2 -> ( ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) -> ( f (Walks ` G ) p -> ( f (Walks ` G ) W /\ ( # ` f ) = 2 ) ) ) ) ) )
83	82	com25 99	. . . . . . . . . . . 12 |- ( f (Walks ` G ) p -> ( f (Walks ` G ) p -> ( ( # ` f ) = 2 -> ( ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) -> ( ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ W = <" a b c "> ) -> ( f (Walks ` G ) W /\ ( # ` f ) = 2 ) ) ) ) ) )
84	83	pm2.43i 52	. . . . . . . . . . 11 |- ( f (Walks ` G ) p -> ( ( # ` f ) = 2 -> ( ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) -> ( ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ W = <" a b c "> ) -> ( f (Walks ` G ) W /\ ( # ` f ) = 2 ) ) ) ) )
85	84	3imp 1256	. . . . . . . . . 10 |- ( ( f (Walks ` G ) p /\ ( # ` f ) = 2 /\ ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) ) -> ( ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ W = <" a b c "> ) -> ( f (Walks ` G ) W /\ ( # ` f ) = 2 ) ) )
86	85	com12 32	. . . . . . . . 9 |- ( ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ W = <" a b c "> ) -> ( ( f (Walks ` G ) p /\ ( # ` f ) = 2 /\ ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) ) -> ( f (Walks ` G ) W /\ ( # ` f ) = 2 ) ) )
87	86	exlimdv 1861	. . . . . . . 8 |- ( ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ W = <" a b c "> ) -> ( E. p ( f (Walks ` G ) p /\ ( # ` f ) = 2 /\ ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) ) -> ( f (Walks ` G ) W /\ ( # ` f ) = 2 ) ) )
88	43, 87	impbid 202	. . . . . . 7 |- ( ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ W = <" a b c "> ) -> ( ( f (Walks ` G ) W /\ ( # ` f ) = 2 ) <-> E. p ( f (Walks ` G ) p /\ ( # ` f ) = 2 /\ ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) ) ) )
89	88	exbidv 1850	. . . . . 6 |- ( ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) /\ W = <" a b c "> ) -> ( E. f ( f (Walks ` G ) W /\ ( # ` f ) = 2 ) <-> E. f E. p ( f (Walks ` G ) p /\ ( # ` f ) = 2 /\ ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) ) ) )
90	89	pm5.32da 673	. . . . 5 |- ( ( ( G e. UPGraph /\ a e. V ) /\ ( b e. V /\ c e. V ) ) -> ( ( W = <" a b c "> /\ E. f ( f (Walks ` G ) W /\ ( # ` f ) = 2 ) ) <-> ( W = <" a b c "> /\ E. f E. p ( f (Walks ` G ) p /\ ( # ` f ) = 2 /\ ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) ) ) ) )
91	90	2rexbidva 3056	. . . 4 |- ( ( G e. UPGraph /\ a e. V ) -> ( E. b e. V E. c e. V ( W = <" a b c "> /\ E. f ( f (Walks ` G ) W /\ ( # ` f ) = 2 ) ) <-> E. b e. V E. c e. V ( W = <" a b c "> /\ E. f E. p ( f (Walks ` G ) p /\ ( # ` f ) = 2 /\ ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) ) ) ) )
92	8, 91	syl5bb 272	. . 3 |- ( ( G e. UPGraph /\ a e. V ) -> ( E. c e. V E. b e. V ( W = <" a b c "> /\ E. f ( f (Walks ` G ) W /\ ( # ` f ) = 2 ) ) <-> E. b e. V E. c e. V ( W = <" a b c "> /\ E. f E. p ( f (Walks ` G ) p /\ ( # ` f ) = 2 /\ ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) ) ) ) )
93	92	rexbidva 3049	. 2 |- ( G e. UPGraph -> ( E. a e. V E. c e. V E. b e. V ( W = <" a b c "> /\ E. f ( f (Walks ` G ) W /\ ( # ` f ) = 2 ) ) <-> E. a e. V E. b e. V E. c e. V ( W = <" a b c "> /\ E. f E. p ( f (Walks ` G ) p /\ ( # ` f ) = 2 /\ ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) ) ) ) )
94	4, 7, 93	3bitrd 294	1 |- ( G e. UPGraph -> ( W e. ( 2 WWalksN G ) <-> E. a e. V E. b e. V E. c e. V ( W = <" a b c "> /\ E. f E. p ( f (Walks ` G ) p /\ ( # ` f ) = 2 /\ ( a = ( p ` 0 ) /\ b = ( p ` 1 ) /\ c = ( p ` 2 ) ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   E.wrex 2913   _Vcvv 3200   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   2c2 11070   3c3 11071   NN0cn0 11292   #chash 13117  Word cword 13291   <"cs3 13587  Vtxcvtx 25874   UPGraph cupgr 25975  Walkscwlks 26492   WWalksN cwwlksn 26718   WWalksNOn cwwlksnon 26719

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-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-wlks 26495  df-wwlks 26722  df-wwlksn 26723  df-wwlksnon 26724

This theorem is referenced by: (None)