Theorem umgrwwlks2on 26850

Description: A walk of length 2 between two vertices as word in a multigraph. This theorem would also hold for pseudographs, but to prove this the cases A = B and/or B = C must be considered separately. (Contributed by Alexander van der Vekens, 18-Feb-2018.) (Revised by AV, 12-May-2021.)

Hypotheses

Ref	Expression
s3wwlks2on.v	|- V = (Vtx ` G )
usgrwwlks2on.e	|- E = (Edg ` G )

Assertion

Ref	Expression
umgrwwlks2on	|- ( ( G e. UMGraph /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( <" A B C "> e. ( A ( 2 WWalksNOn G ) C ) <-> ( { A , B } e. E /\ { B , C } e. E ) ) )

Proof of Theorem umgrwwlks2on

Dummy variables f p are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		umgrupgr 25998	. . . 4 |- ( G e. UMGraph -> G e. UPGraph )
2	1	adantr 481	. . 3 |- ( ( G e. UMGraph /\ ( A e. V /\ B e. V /\ C e. V ) ) -> G e. UPGraph )
3		simp1 1061	. . . 4 |- ( ( A e. V /\ B e. V /\ C e. V ) -> A e. V )
4	3	adantl 482	. . 3 |- ( ( G e. UMGraph /\ ( A e. V /\ B e. V /\ C e. V ) ) -> A e. V )
5		simpr3 1069	. . 3 |- ( ( G e. UMGraph /\ ( A e. V /\ B e. V /\ C e. V ) ) -> C e. V )
6		s3wwlks2on.v	. . . 4 |- V = (Vtx ` G )
7	6	s3wwlks2on 26849	. . 3 |- ( ( G e. UPGraph /\ A e. V /\ C e. V ) -> ( <" A B C "> e. ( A ( 2 WWalksNOn G ) C ) <-> E. f ( f (Walks ` G ) <" A B C "> /\ ( # ` f ) = 2 ) ) )
8	2, 4, 5, 7	syl3anc 1326	. 2 |- ( ( G e. UMGraph /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( <" A B C "> e. ( A ( 2 WWalksNOn G ) C ) <-> E. f ( f (Walks ` G ) <" A B C "> /\ ( # ` f ) = 2 ) ) )
9		eqid 2622	. . . . . . . 8 |- (iEdg ` G ) = (iEdg ` G )
10	6, 9	upgr2wlk 26564	. . . . . . 7 |- ( G e. UPGraph -> ( ( f (Walks ` G ) <" A B C "> /\ ( # ` f ) = 2 ) <-> ( f : ( 0..^ 2 ) --> dom (iEdg ` G ) /\ <" A B C "> : ( 0 ... 2 ) --> V /\ ( ( (iEdg ` G ) ` ( f ` 0 ) ) = { ( <" A B C "> ` 0 ) , ( <" A B C "> ` 1 ) } /\ ( (iEdg ` G ) ` ( f ` 1 ) ) = { ( <" A B C "> ` 1 ) , ( <" A B C "> ` 2 ) } ) ) ) )
11	1, 10	syl 17	. . . . . 6 |- ( G e. UMGraph -> ( ( f (Walks ` G ) <" A B C "> /\ ( # ` f ) = 2 ) <-> ( f : ( 0..^ 2 ) --> dom (iEdg ` G ) /\ <" A B C "> : ( 0 ... 2 ) --> V /\ ( ( (iEdg ` G ) ` ( f ` 0 ) ) = { ( <" A B C "> ` 0 ) , ( <" A B C "> ` 1 ) } /\ ( (iEdg ` G ) ` ( f ` 1 ) ) = { ( <" A B C "> ` 1 ) , ( <" A B C "> ` 2 ) } ) ) ) )
12	11	adantr 481	. . . . 5 |- ( ( G e. UMGraph /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( ( f (Walks ` G ) <" A B C "> /\ ( # ` f ) = 2 ) <-> ( f : ( 0..^ 2 ) --> dom (iEdg ` G ) /\ <" A B C "> : ( 0 ... 2 ) --> V /\ ( ( (iEdg ` G ) ` ( f ` 0 ) ) = { ( <" A B C "> ` 0 ) , ( <" A B C "> ` 1 ) } /\ ( (iEdg ` G ) ` ( f ` 1 ) ) = { ( <" A B C "> ` 1 ) , ( <" A B C "> ` 2 ) } ) ) ) )
13		s3fv0 13636	. . . . . . . . . . . 12 |- ( A e. V -> ( <" A B C "> ` 0 ) = A )
14	13	3ad2ant1 1082	. . . . . . . . . . 11 |- ( ( A e. V /\ B e. V /\ C e. V ) -> ( <" A B C "> ` 0 ) = A )
15		s3fv1 13637	. . . . . . . . . . . 12 |- ( B e. V -> ( <" A B C "> ` 1 ) = B )
16	15	3ad2ant2 1083	. . . . . . . . . . 11 |- ( ( A e. V /\ B e. V /\ C e. V ) -> ( <" A B C "> ` 1 ) = B )
17	14, 16	preq12d 4276	. . . . . . . . . 10 |- ( ( A e. V /\ B e. V /\ C e. V ) -> { ( <" A B C "> ` 0 ) , ( <" A B C "> ` 1 ) } = { A , B } )
18	17	eqeq2d 2632	. . . . . . . . 9 |- ( ( A e. V /\ B e. V /\ C e. V ) -> ( ( (iEdg ` G ) ` ( f ` 0 ) ) = { ( <" A B C "> ` 0 ) , ( <" A B C "> ` 1 ) } <-> ( (iEdg ` G ) ` ( f ` 0 ) ) = { A , B } ) )
19		s3fv2 13638	. . . . . . . . . . . 12 |- ( C e. V -> ( <" A B C "> ` 2 ) = C )
20	19	3ad2ant3 1084	. . . . . . . . . . 11 |- ( ( A e. V /\ B e. V /\ C e. V ) -> ( <" A B C "> ` 2 ) = C )
21	16, 20	preq12d 4276	. . . . . . . . . 10 |- ( ( A e. V /\ B e. V /\ C e. V ) -> { ( <" A B C "> ` 1 ) , ( <" A B C "> ` 2 ) } = { B , C } )
22	21	eqeq2d 2632	. . . . . . . . 9 |- ( ( A e. V /\ B e. V /\ C e. V ) -> ( ( (iEdg ` G ) ` ( f ` 1 ) ) = { ( <" A B C "> ` 1 ) , ( <" A B C "> ` 2 ) } <-> ( (iEdg ` G ) ` ( f ` 1 ) ) = { B , C } ) )
23	18, 22	anbi12d 747	. . . . . . . 8 |- ( ( A e. V /\ B e. V /\ C e. V ) -> ( ( ( (iEdg ` G ) ` ( f ` 0 ) ) = { ( <" A B C "> ` 0 ) , ( <" A B C "> ` 1 ) } /\ ( (iEdg ` G ) ` ( f ` 1 ) ) = { ( <" A B C "> ` 1 ) , ( <" A B C "> ` 2 ) } ) <-> ( ( (iEdg ` G ) ` ( f ` 0 ) ) = { A , B } /\ ( (iEdg ` G ) ` ( f ` 1 ) ) = { B , C } ) ) )
24	23	adantl 482	. . . . . . 7 |- ( ( G e. UMGraph /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( ( ( (iEdg ` G ) ` ( f ` 0 ) ) = { ( <" A B C "> ` 0 ) , ( <" A B C "> ` 1 ) } /\ ( (iEdg ` G ) ` ( f ` 1 ) ) = { ( <" A B C "> ` 1 ) , ( <" A B C "> ` 2 ) } ) <-> ( ( (iEdg ` G ) ` ( f ` 0 ) ) = { A , B } /\ ( (iEdg ` G ) ` ( f ` 1 ) ) = { B , C } ) ) )
25	24	3anbi3d 1405	. . . . . 6 |- ( ( G e. UMGraph /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( ( f : ( 0..^ 2 ) --> dom (iEdg ` G ) /\ <" A B C "> : ( 0 ... 2 ) --> V /\ ( ( (iEdg ` G ) ` ( f ` 0 ) ) = { ( <" A B C "> ` 0 ) , ( <" A B C "> ` 1 ) } /\ ( (iEdg ` G ) ` ( f ` 1 ) ) = { ( <" A B C "> ` 1 ) , ( <" A B C "> ` 2 ) } ) ) <-> ( f : ( 0..^ 2 ) --> dom (iEdg ` G ) /\ <" A B C "> : ( 0 ... 2 ) --> V /\ ( ( (iEdg ` G ) ` ( f ` 0 ) ) = { A , B } /\ ( (iEdg ` G ) ` ( f ` 1 ) ) = { B , C } ) ) ) )
26		umgruhgr 25999	. . . . . . . . . . 11 |- ( G e. UMGraph -> G e. UHGraph )
27	9	uhgrfun 25961	. . . . . . . . . . 11 |- ( G e. UHGraph -> Fun (iEdg ` G ) )
28		fdmrn 6064	. . . . . . . . . . . 12 |- ( Fun (iEdg ` G ) <-> (iEdg ` G ) : dom (iEdg ` G ) --> ran (iEdg ` G ) )
29		simpr 477	. . . . . . . . . . . . . . . . 17 |- ( ( f : ( 0..^ 2 ) --> dom (iEdg ` G ) /\ (iEdg ` G ) : dom (iEdg ` G ) --> ran (iEdg ` G ) ) -> (iEdg ` G ) : dom (iEdg ` G ) --> ran (iEdg ` G ) )
30		id 22	. . . . . . . . . . . . . . . . . . 19 |- ( f : ( 0..^ 2 ) --> dom (iEdg ` G ) -> f : ( 0..^ 2 ) --> dom (iEdg ` G ) )
31		c0ex 10034	. . . . . . . . . . . . . . . . . . . . . 22 |- 0 e. _V
32	31	prid1 4297	. . . . . . . . . . . . . . . . . . . . 21 |- 0 e. { 0 , 1 }
33		fzo0to2pr 12553	. . . . . . . . . . . . . . . . . . . . 21 |- ( 0..^ 2 ) = { 0 , 1 }
34	32, 33	eleqtrri 2700	. . . . . . . . . . . . . . . . . . . 20 |- 0 e. ( 0..^ 2 )
35	34	a1i 11	. . . . . . . . . . . . . . . . . . 19 |- ( f : ( 0..^ 2 ) --> dom (iEdg ` G ) -> 0 e. ( 0..^ 2 ) )
36	30, 35	ffvelrnd 6360	. . . . . . . . . . . . . . . . . 18 |- ( f : ( 0..^ 2 ) --> dom (iEdg ` G ) -> ( f ` 0 ) e. dom (iEdg ` G ) )
37	36	adantr 481	. . . . . . . . . . . . . . . . 17 |- ( ( f : ( 0..^ 2 ) --> dom (iEdg ` G ) /\ (iEdg ` G ) : dom (iEdg ` G ) --> ran (iEdg ` G ) ) -> ( f ` 0 ) e. dom (iEdg ` G ) )
38	29, 37	ffvelrnd 6360	. . . . . . . . . . . . . . . 16 |- ( ( f : ( 0..^ 2 ) --> dom (iEdg ` G ) /\ (iEdg ` G ) : dom (iEdg ` G ) --> ran (iEdg ` G ) ) -> ( (iEdg ` G ) ` ( f ` 0 ) ) e. ran (iEdg ` G ) )
39		1ex 10035	. . . . . . . . . . . . . . . . . . . . . 22 |- 1 e. _V
40	39	prid2 4298	. . . . . . . . . . . . . . . . . . . . 21 |- 1 e. { 0 , 1 }
41	40, 33	eleqtrri 2700	. . . . . . . . . . . . . . . . . . . 20 |- 1 e. ( 0..^ 2 )
42	41	a1i 11	. . . . . . . . . . . . . . . . . . 19 |- ( f : ( 0..^ 2 ) --> dom (iEdg ` G ) -> 1 e. ( 0..^ 2 ) )
43	30, 42	ffvelrnd 6360	. . . . . . . . . . . . . . . . . 18 |- ( f : ( 0..^ 2 ) --> dom (iEdg ` G ) -> ( f ` 1 ) e. dom (iEdg ` G ) )
44	43	adantr 481	. . . . . . . . . . . . . . . . 17 |- ( ( f : ( 0..^ 2 ) --> dom (iEdg ` G ) /\ (iEdg ` G ) : dom (iEdg ` G ) --> ran (iEdg ` G ) ) -> ( f ` 1 ) e. dom (iEdg ` G ) )
45	29, 44	ffvelrnd 6360	. . . . . . . . . . . . . . . 16 |- ( ( f : ( 0..^ 2 ) --> dom (iEdg ` G ) /\ (iEdg ` G ) : dom (iEdg ` G ) --> ran (iEdg ` G ) ) -> ( (iEdg ` G ) ` ( f ` 1 ) ) e. ran (iEdg ` G ) )
46	38, 45	jca 554	. . . . . . . . . . . . . . 15 |- ( ( f : ( 0..^ 2 ) --> dom (iEdg ` G ) /\ (iEdg ` G ) : dom (iEdg ` G ) --> ran (iEdg ` G ) ) -> ( ( (iEdg ` G ) ` ( f ` 0 ) ) e. ran (iEdg ` G ) /\ ( (iEdg ` G ) ` ( f ` 1 ) ) e. ran (iEdg ` G ) ) )
47	46	ex 450	. . . . . . . . . . . . . 14 |- ( f : ( 0..^ 2 ) --> dom (iEdg ` G ) -> ( (iEdg ` G ) : dom (iEdg ` G ) --> ran (iEdg ` G ) -> ( ( (iEdg ` G ) ` ( f ` 0 ) ) e. ran (iEdg ` G ) /\ ( (iEdg ` G ) ` ( f ` 1 ) ) e. ran (iEdg ` G ) ) ) )
48	47	3ad2ant1 1082	. . . . . . . . . . . . 13 |- ( ( f : ( 0..^ 2 ) --> dom (iEdg ` G ) /\ <" A B C "> : ( 0 ... 2 ) --> V /\ ( ( (iEdg ` G ) ` ( f ` 0 ) ) = { A , B } /\ ( (iEdg ` G ) ` ( f ` 1 ) ) = { B , C } ) ) -> ( (iEdg ` G ) : dom (iEdg ` G ) --> ran (iEdg ` G ) -> ( ( (iEdg ` G ) ` ( f ` 0 ) ) e. ran (iEdg ` G ) /\ ( (iEdg ` G ) ` ( f ` 1 ) ) e. ran (iEdg ` G ) ) ) )
49	48	com12 32	. . . . . . . . . . . 12 |- ( (iEdg ` G ) : dom (iEdg ` G ) --> ran (iEdg ` G ) -> ( ( f : ( 0..^ 2 ) --> dom (iEdg ` G ) /\ <" A B C "> : ( 0 ... 2 ) --> V /\ ( ( (iEdg ` G ) ` ( f ` 0 ) ) = { A , B } /\ ( (iEdg ` G ) ` ( f ` 1 ) ) = { B , C } ) ) -> ( ( (iEdg ` G ) ` ( f ` 0 ) ) e. ran (iEdg ` G ) /\ ( (iEdg ` G ) ` ( f ` 1 ) ) e. ran (iEdg ` G ) ) ) )
50	28, 49	sylbi 207	. . . . . . . . . . 11 |- ( Fun (iEdg ` G ) -> ( ( f : ( 0..^ 2 ) --> dom (iEdg ` G ) /\ <" A B C "> : ( 0 ... 2 ) --> V /\ ( ( (iEdg ` G ) ` ( f ` 0 ) ) = { A , B } /\ ( (iEdg ` G ) ` ( f ` 1 ) ) = { B , C } ) ) -> ( ( (iEdg ` G ) ` ( f ` 0 ) ) e. ran (iEdg ` G ) /\ ( (iEdg ` G ) ` ( f ` 1 ) ) e. ran (iEdg ` G ) ) ) )
51	26, 27, 50	3syl 18	. . . . . . . . . 10 |- ( G e. UMGraph -> ( ( f : ( 0..^ 2 ) --> dom (iEdg ` G ) /\ <" A B C "> : ( 0 ... 2 ) --> V /\ ( ( (iEdg ` G ) ` ( f ` 0 ) ) = { A , B } /\ ( (iEdg ` G ) ` ( f ` 1 ) ) = { B , C } ) ) -> ( ( (iEdg ` G ) ` ( f ` 0 ) ) e. ran (iEdg ` G ) /\ ( (iEdg ` G ) ` ( f ` 1 ) ) e. ran (iEdg ` G ) ) ) )
52	51	imp 445	. . . . . . . . 9 |- ( ( G e. UMGraph /\ ( f : ( 0..^ 2 ) --> dom (iEdg ` G ) /\ <" A B C "> : ( 0 ... 2 ) --> V /\ ( ( (iEdg ` G ) ` ( f ` 0 ) ) = { A , B } /\ ( (iEdg ` G ) ` ( f ` 1 ) ) = { B , C } ) ) ) -> ( ( (iEdg ` G ) ` ( f ` 0 ) ) e. ran (iEdg ` G ) /\ ( (iEdg ` G ) ` ( f ` 1 ) ) e. ran (iEdg ` G ) ) )
53		eqcom 2629	. . . . . . . . . . . . . . 15 |- ( ( (iEdg ` G ) ` ( f ` 0 ) ) = { A , B } <-> { A , B } = ( (iEdg ` G ) ` ( f ` 0 ) ) )
54	53	biimpi 206	. . . . . . . . . . . . . 14 |- ( ( (iEdg ` G ) ` ( f ` 0 ) ) = { A , B } -> { A , B } = ( (iEdg ` G ) ` ( f ` 0 ) ) )
55	54	adantr 481	. . . . . . . . . . . . 13 |- ( ( ( (iEdg ` G ) ` ( f ` 0 ) ) = { A , B } /\ ( (iEdg ` G ) ` ( f ` 1 ) ) = { B , C } ) -> { A , B } = ( (iEdg ` G ) ` ( f ` 0 ) ) )
56	55	3ad2ant3 1084	. . . . . . . . . . . 12 |- ( ( f : ( 0..^ 2 ) --> dom (iEdg ` G ) /\ <" A B C "> : ( 0 ... 2 ) --> V /\ ( ( (iEdg ` G ) ` ( f ` 0 ) ) = { A , B } /\ ( (iEdg ` G ) ` ( f ` 1 ) ) = { B , C } ) ) -> { A , B } = ( (iEdg ` G ) ` ( f ` 0 ) ) )
57	56	adantl 482	. . . . . . . . . . 11 |- ( ( G e. UMGraph /\ ( f : ( 0..^ 2 ) --> dom (iEdg ` G ) /\ <" A B C "> : ( 0 ... 2 ) --> V /\ ( ( (iEdg ` G ) ` ( f ` 0 ) ) = { A , B } /\ ( (iEdg ` G ) ` ( f ` 1 ) ) = { B , C } ) ) ) -> { A , B } = ( (iEdg ` G ) ` ( f ` 0 ) ) )
58		usgrwwlks2on.e	. . . . . . . . . . . . 13 |- E = (Edg ` G )
59		edgval 25941	. . . . . . . . . . . . 13 |- (Edg ` G ) = ran (iEdg ` G )
60	58, 59	eqtri 2644	. . . . . . . . . . . 12 |- E = ran (iEdg ` G )
61	60	a1i 11	. . . . . . . . . . 11 |- ( ( G e. UMGraph /\ ( f : ( 0..^ 2 ) --> dom (iEdg ` G ) /\ <" A B C "> : ( 0 ... 2 ) --> V /\ ( ( (iEdg ` G ) ` ( f ` 0 ) ) = { A , B } /\ ( (iEdg ` G ) ` ( f ` 1 ) ) = { B , C } ) ) ) -> E = ran (iEdg ` G ) )
62	57, 61	eleq12d 2695	. . . . . . . . . 10 |- ( ( G e. UMGraph /\ ( f : ( 0..^ 2 ) --> dom (iEdg ` G ) /\ <" A B C "> : ( 0 ... 2 ) --> V /\ ( ( (iEdg ` G ) ` ( f ` 0 ) ) = { A , B } /\ ( (iEdg ` G ) ` ( f ` 1 ) ) = { B , C } ) ) ) -> ( { A , B } e. E <-> ( (iEdg ` G ) ` ( f ` 0 ) ) e. ran (iEdg ` G ) ) )
63		eqcom 2629	. . . . . . . . . . . . . . 15 |- ( ( (iEdg ` G ) ` ( f ` 1 ) ) = { B , C } <-> { B , C } = ( (iEdg ` G ) ` ( f ` 1 ) ) )
64	63	biimpi 206	. . . . . . . . . . . . . 14 |- ( ( (iEdg ` G ) ` ( f ` 1 ) ) = { B , C } -> { B , C } = ( (iEdg ` G ) ` ( f ` 1 ) ) )
65	64	adantl 482	. . . . . . . . . . . . 13 |- ( ( ( (iEdg ` G ) ` ( f ` 0 ) ) = { A , B } /\ ( (iEdg ` G ) ` ( f ` 1 ) ) = { B , C } ) -> { B , C } = ( (iEdg ` G ) ` ( f ` 1 ) ) )
66	65	3ad2ant3 1084	. . . . . . . . . . . 12 |- ( ( f : ( 0..^ 2 ) --> dom (iEdg ` G ) /\ <" A B C "> : ( 0 ... 2 ) --> V /\ ( ( (iEdg ` G ) ` ( f ` 0 ) ) = { A , B } /\ ( (iEdg ` G ) ` ( f ` 1 ) ) = { B , C } ) ) -> { B , C } = ( (iEdg ` G ) ` ( f ` 1 ) ) )
67	66	adantl 482	. . . . . . . . . . 11 |- ( ( G e. UMGraph /\ ( f : ( 0..^ 2 ) --> dom (iEdg ` G ) /\ <" A B C "> : ( 0 ... 2 ) --> V /\ ( ( (iEdg ` G ) ` ( f ` 0 ) ) = { A , B } /\ ( (iEdg ` G ) ` ( f ` 1 ) ) = { B , C } ) ) ) -> { B , C } = ( (iEdg ` G ) ` ( f ` 1 ) ) )
68	67, 61	eleq12d 2695	. . . . . . . . . 10 |- ( ( G e. UMGraph /\ ( f : ( 0..^ 2 ) --> dom (iEdg ` G ) /\ <" A B C "> : ( 0 ... 2 ) --> V /\ ( ( (iEdg ` G ) ` ( f ` 0 ) ) = { A , B } /\ ( (iEdg ` G ) ` ( f ` 1 ) ) = { B , C } ) ) ) -> ( { B , C } e. E <-> ( (iEdg ` G ) ` ( f ` 1 ) ) e. ran (iEdg ` G ) ) )
69	62, 68	anbi12d 747	. . . . . . . . 9 |- ( ( G e. UMGraph /\ ( f : ( 0..^ 2 ) --> dom (iEdg ` G ) /\ <" A B C "> : ( 0 ... 2 ) --> V /\ ( ( (iEdg ` G ) ` ( f ` 0 ) ) = { A , B } /\ ( (iEdg ` G ) ` ( f ` 1 ) ) = { B , C } ) ) ) -> ( ( { A , B } e. E /\ { B , C } e. E ) <-> ( ( (iEdg ` G ) ` ( f ` 0 ) ) e. ran (iEdg ` G ) /\ ( (iEdg ` G ) ` ( f ` 1 ) ) e. ran (iEdg ` G ) ) ) )
70	52, 69	mpbird 247	. . . . . . . 8 |- ( ( G e. UMGraph /\ ( f : ( 0..^ 2 ) --> dom (iEdg ` G ) /\ <" A B C "> : ( 0 ... 2 ) --> V /\ ( ( (iEdg ` G ) ` ( f ` 0 ) ) = { A , B } /\ ( (iEdg ` G ) ` ( f ` 1 ) ) = { B , C } ) ) ) -> ( { A , B } e. E /\ { B , C } e. E ) )
71	70	ex 450	. . . . . . 7 |- ( G e. UMGraph -> ( ( f : ( 0..^ 2 ) --> dom (iEdg ` G ) /\ <" A B C "> : ( 0 ... 2 ) --> V /\ ( ( (iEdg ` G ) ` ( f ` 0 ) ) = { A , B } /\ ( (iEdg ` G ) ` ( f ` 1 ) ) = { B , C } ) ) -> ( { A , B } e. E /\ { B , C } e. E ) ) )
72	71	adantr 481	. . . . . 6 |- ( ( G e. UMGraph /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( ( f : ( 0..^ 2 ) --> dom (iEdg ` G ) /\ <" A B C "> : ( 0 ... 2 ) --> V /\ ( ( (iEdg ` G ) ` ( f ` 0 ) ) = { A , B } /\ ( (iEdg ` G ) ` ( f ` 1 ) ) = { B , C } ) ) -> ( { A , B } e. E /\ { B , C } e. E ) ) )
73	25, 72	sylbid 230	. . . . 5 |- ( ( G e. UMGraph /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( ( f : ( 0..^ 2 ) --> dom (iEdg ` G ) /\ <" A B C "> : ( 0 ... 2 ) --> V /\ ( ( (iEdg ` G ) ` ( f ` 0 ) ) = { ( <" A B C "> ` 0 ) , ( <" A B C "> ` 1 ) } /\ ( (iEdg ` G ) ` ( f ` 1 ) ) = { ( <" A B C "> ` 1 ) , ( <" A B C "> ` 2 ) } ) ) -> ( { A , B } e. E /\ { B , C } e. E ) ) )
74	12, 73	sylbid 230	. . . 4 |- ( ( G e. UMGraph /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( ( f (Walks ` G ) <" A B C "> /\ ( # ` f ) = 2 ) -> ( { A , B } e. E /\ { B , C } e. E ) ) )
75	74	exlimdv 1861	. . 3 |- ( ( G e. UMGraph /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( E. f ( f (Walks ` G ) <" A B C "> /\ ( # ` f ) = 2 ) -> ( { A , B } e. E /\ { B , C } e. E ) ) )
76	58	umgr2wlk 26845	. . . . . . 7 |- ( ( G e. UMGraph /\ { A , B } e. E /\ { B , C } e. E ) -> E. f E. p ( f (Walks ` G ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) )
77		wlklenvp1 26514	. . . . . . . . . . . . . . . . . . . 20 |- ( f (Walks ` G ) p -> ( # ` p ) = ( ( # ` f ) + 1 ) )
78		oveq1 6657	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( # ` f ) = 2 -> ( ( # ` f ) + 1 ) = ( 2 + 1 ) )
79		2p1e3 11151	. . . . . . . . . . . . . . . . . . . . . 22 |- ( 2 + 1 ) = 3
80	78, 79	syl6eq 2672	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( # ` f ) = 2 -> ( ( # ` f ) + 1 ) = 3 )
81	80	adantr 481	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) -> ( ( # ` f ) + 1 ) = 3 )
82	77, 81	sylan9eq 2676	. . . . . . . . . . . . . . . . . . 19 |- ( ( f (Walks ` G ) p /\ ( ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) -> ( # ` p ) = 3 )
83		eqcom 2629	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( A = ( p ` 0 ) <-> ( p ` 0 ) = A )
84		eqcom 2629	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( B = ( p ` 1 ) <-> ( p ` 1 ) = B )
85		eqcom 2629	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( C = ( p ` 2 ) <-> ( p ` 2 ) = C )
86	83, 84, 85	3anbi123i 1251	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) <-> ( ( p ` 0 ) = A /\ ( p ` 1 ) = B /\ ( p ` 2 ) = C ) )
87	86	biimpi 206	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) -> ( ( p ` 0 ) = A /\ ( p ` 1 ) = B /\ ( p ` 2 ) = C ) )
88	87	adantl 482	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) -> ( ( p ` 0 ) = A /\ ( p ` 1 ) = B /\ ( p ` 2 ) = C ) )
89	88	adantl 482	. . . . . . . . . . . . . . . . . . 19 |- ( ( f (Walks ` G ) p /\ ( ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) -> ( ( p ` 0 ) = A /\ ( p ` 1 ) = B /\ ( p ` 2 ) = C ) )
90	82, 89	jca 554	. . . . . . . . . . . . . . . . . 18 |- ( ( f (Walks ` G ) p /\ ( ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) -> ( ( # ` p ) = 3 /\ ( ( p ` 0 ) = A /\ ( p ` 1 ) = B /\ ( p ` 2 ) = C ) ) )
91	6	wlkpwrd 26513	. . . . . . . . . . . . . . . . . . . . 21 |- ( f (Walks ` G ) p -> p e. Word V )
92	80	eqeq2d 2632	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( # ` f ) = 2 -> ( ( # ` p ) = ( ( # ` f ) + 1 ) <-> ( # ` p ) = 3 ) )
93	92	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( p e. Word V /\ ( # ` f ) = 2 ) -> ( ( # ` p ) = ( ( # ` f ) + 1 ) <-> ( # ` p ) = 3 ) )
94		simp1 1061	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( p e. Word V /\ ( # ` p ) = 3 /\ ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) -> p e. Word V )
95		oveq2 6658	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- ( ( # ` p ) = 3 -> ( 0..^ ( # ` p ) ) = ( 0..^ 3 ) )
96		fzo0to3tp 12554	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- ( 0..^ 3 ) = { 0 , 1 , 2 }
97	95, 96	syl6eq 2672	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- ( ( # ` p ) = 3 -> ( 0..^ ( # ` p ) ) = { 0 , 1 , 2 } )
98	31	tpid1 4303	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 |- 0 e. { 0 , 1 , 2 }
99		eleq2 2690	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 |- ( ( 0..^ ( # ` p ) ) = { 0 , 1 , 2 } -> ( 0 e. ( 0..^ ( # ` p ) ) <-> 0 e. { 0 , 1 , 2 } ) )
100	98, 99	mpbiri 248	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- ( ( 0..^ ( # ` p ) ) = { 0 , 1 , 2 } -> 0 e. ( 0..^ ( # ` p ) ) )
101		wrdsymbcl 13318	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- ( ( p e. Word V /\ 0 e. ( 0..^ ( # ` p ) ) ) -> ( p ` 0 ) e. V )
102	100, 101	sylan2 491	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- ( ( p e. Word V /\ ( 0..^ ( # ` p ) ) = { 0 , 1 , 2 } ) -> ( p ` 0 ) e. V )
103	39	tpid2 4304	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 |- 1 e. { 0 , 1 , 2 }
104		eleq2 2690	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 |- ( ( 0..^ ( # ` p ) ) = { 0 , 1 , 2 } -> ( 1 e. ( 0..^ ( # ` p ) ) <-> 1 e. { 0 , 1 , 2 } ) )
105	103, 104	mpbiri 248	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- ( ( 0..^ ( # ` p ) ) = { 0 , 1 , 2 } -> 1 e. ( 0..^ ( # ` p ) ) )
106		wrdsymbcl 13318	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- ( ( p e. Word V /\ 1 e. ( 0..^ ( # ` p ) ) ) -> ( p ` 1 ) e. V )
107	105, 106	sylan2 491	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- ( ( p e. Word V /\ ( 0..^ ( # ` p ) ) = { 0 , 1 , 2 } ) -> ( p ` 1 ) e. V )
108		2ex 11092	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 |- 2 e. _V
109	108	tpid3 4307	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 |- 2 e. { 0 , 1 , 2 }
110		eleq2 2690	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 |- ( ( 0..^ ( # ` p ) ) = { 0 , 1 , 2 } -> ( 2 e. ( 0..^ ( # ` p ) ) <-> 2 e. { 0 , 1 , 2 } ) )
111	109, 110	mpbiri 248	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- ( ( 0..^ ( # ` p ) ) = { 0 , 1 , 2 } -> 2 e. ( 0..^ ( # ` p ) ) )
112		wrdsymbcl 13318	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- ( ( p e. Word V /\ 2 e. ( 0..^ ( # ` p ) ) ) -> ( p ` 2 ) e. V )
113	111, 112	sylan2 491	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- ( ( p e. Word V /\ ( 0..^ ( # ` p ) ) = { 0 , 1 , 2 } ) -> ( p ` 2 ) e. V )
114	102, 107, 113	3jca 1242	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- ( ( p e. Word V /\ ( 0..^ ( # ` p ) ) = { 0 , 1 , 2 } ) -> ( ( p ` 0 ) e. V /\ ( p ` 1 ) e. V /\ ( p ` 2 ) e. V ) )
115	97, 114	sylan2 491	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ( ( p e. Word V /\ ( # ` p ) = 3 ) -> ( ( p ` 0 ) e. V /\ ( p ` 1 ) e. V /\ ( p ` 2 ) e. V ) )
116	115	3adant3 1081	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( ( p e. Word V /\ ( # ` p ) = 3 /\ ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) -> ( ( p ` 0 ) e. V /\ ( p ` 1 ) e. V /\ ( p ` 2 ) e. V ) )
117		eleq1 2689	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- ( A = ( p ` 0 ) -> ( A e. V <-> ( p ` 0 ) e. V ) )
118	117	3ad2ant1 1082	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- ( ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) -> ( A e. V <-> ( p ` 0 ) e. V ) )
119		eleq1 2689	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- ( B = ( p ` 1 ) -> ( B e. V <-> ( p ` 1 ) e. V ) )
120	119	3ad2ant2 1083	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- ( ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) -> ( B e. V <-> ( p ` 1 ) e. V ) )
121		eleq1 2689	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- ( C = ( p ` 2 ) -> ( C e. V <-> ( p ` 2 ) e. V ) )
122	121	3ad2ant3 1084	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- ( ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) -> ( C e. V <-> ( p ` 2 ) e. V ) )
123	118, 120, 122	3anbi123d 1399	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ( ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) -> ( ( A e. V /\ B e. V /\ C e. V ) <-> ( ( p ` 0 ) e. V /\ ( p ` 1 ) e. V /\ ( p ` 2 ) e. V ) ) )
124	123	3ad2ant3 1084	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( ( p e. Word V /\ ( # ` p ) = 3 /\ ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) -> ( ( A e. V /\ B e. V /\ C e. V ) <-> ( ( p ` 0 ) e. V /\ ( p ` 1 ) e. V /\ ( p ` 2 ) e. V ) ) )
125	116, 124	mpbird 247	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( p e. Word V /\ ( # ` p ) = 3 /\ ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) -> ( A e. V /\ B e. V /\ C e. V ) )
126	94, 125	jca 554	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( p e. Word V /\ ( # ` p ) = 3 /\ ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) -> ( p e. Word V /\ ( A e. V /\ B e. V /\ C e. V ) ) )
127	126	3exp 1264	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( p e. Word V -> ( ( # ` p ) = 3 -> ( ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) -> ( p e. Word V /\ ( A e. V /\ B e. V /\ C e. V ) ) ) ) )
128	127	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( p e. Word V /\ ( # ` f ) = 2 ) -> ( ( # ` p ) = 3 -> ( ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) -> ( p e. Word V /\ ( A e. V /\ B e. V /\ C e. V ) ) ) ) )
129	93, 128	sylbid 230	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( p e. Word V /\ ( # ` f ) = 2 ) -> ( ( # ` p ) = ( ( # ` f ) + 1 ) -> ( ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) -> ( p e. Word V /\ ( A e. V /\ B e. V /\ C e. V ) ) ) ) )
130	129	impancom 456	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( p e. Word V /\ ( # ` p ) = ( ( # ` f ) + 1 ) ) -> ( ( # ` f ) = 2 -> ( ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) -> ( p e. Word V /\ ( A e. V /\ B e. V /\ C e. V ) ) ) ) )
131	130	impd 447	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( p e. Word V /\ ( # ` p ) = ( ( # ` f ) + 1 ) ) -> ( ( ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) -> ( p e. Word V /\ ( A e. V /\ B e. V /\ C e. V ) ) ) )
132	91, 77, 131	syl2anc 693	. . . . . . . . . . . . . . . . . . . 20 |- ( f (Walks ` G ) p -> ( ( ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) -> ( p e. Word V /\ ( A e. V /\ B e. V /\ C e. V ) ) ) )
133	132	imp 445	. . . . . . . . . . . . . . . . . . 19 |- ( ( f (Walks ` G ) p /\ ( ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) -> ( p e. Word V /\ ( A e. V /\ B e. V /\ C e. V ) ) )
134		eqwrds3 13704	. . . . . . . . . . . . . . . . . . 19 |- ( ( 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 ) ) ) )
135	133, 134	syl 17	. . . . . . . . . . . . . . . . . 18 |- ( ( f (Walks ` G ) p /\ ( ( # ` 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 ) ) ) )
136	90, 135	mpbird 247	. . . . . . . . . . . . . . . . 17 |- ( ( f (Walks ` G ) p /\ ( ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) -> p = <" A B C "> )
137	136	breq2d 4665	. . . . . . . . . . . . . . . 16 |- ( ( f (Walks ` G ) p /\ ( ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) -> ( f (Walks ` G ) p <-> f (Walks ` G ) <" A B C "> ) )
138	137	biimpd 219	. . . . . . . . . . . . . . 15 |- ( ( f (Walks ` G ) p /\ ( ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) -> ( f (Walks ` G ) p -> f (Walks ` G ) <" A B C "> ) )
139	138	ex 450	. . . . . . . . . . . . . 14 |- ( f (Walks ` G ) p -> ( ( ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) -> ( f (Walks ` G ) p -> f (Walks ` G ) <" A B C "> ) ) )
140	139	pm2.43a 54	. . . . . . . . . . . . 13 |- ( f (Walks ` G ) p -> ( ( ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) -> f (Walks ` G ) <" A B C "> ) )
141	140	3impib 1262	. . . . . . . . . . . 12 |- ( ( f (Walks ` G ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) -> f (Walks ` G ) <" A B C "> )
142	141	adantl 482	. . . . . . . . . . 11 |- ( ( ( A e. V /\ B e. V /\ C e. V ) /\ ( f (Walks ` G ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) -> f (Walks ` G ) <" A B C "> )
143		simpr2 1068	. . . . . . . . . . 11 |- ( ( ( A e. V /\ B e. V /\ C e. V ) /\ ( f (Walks ` G ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) -> ( # ` f ) = 2 )
144	142, 143	jca 554	. . . . . . . . . 10 |- ( ( ( A e. V /\ B e. V /\ C e. V ) /\ ( f (Walks ` G ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) -> ( f (Walks ` G ) <" A B C "> /\ ( # ` f ) = 2 ) )
145	144	ex 450	. . . . . . . . 9 |- ( ( A e. V /\ B e. V /\ C e. V ) -> ( ( f (Walks ` G ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) -> ( f (Walks ` G ) <" A B C "> /\ ( # ` f ) = 2 ) ) )
146	145	exlimdv 1861	. . . . . . . 8 |- ( ( A e. V /\ B e. V /\ C e. V ) -> ( E. p ( f (Walks ` G ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) -> ( f (Walks ` G ) <" A B C "> /\ ( # ` f ) = 2 ) ) )
147	146	eximdv 1846	. . . . . . 7 |- ( ( A e. V /\ B e. V /\ C e. V ) -> ( E. f E. p ( f (Walks ` G ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) -> E. f ( f (Walks ` G ) <" A B C "> /\ ( # ` f ) = 2 ) ) )
148	76, 147	syl5com 31	. . . . . 6 |- ( ( G e. UMGraph /\ { A , B } e. E /\ { B , C } e. E ) -> ( ( A e. V /\ B e. V /\ C e. V ) -> E. f ( f (Walks ` G ) <" A B C "> /\ ( # ` f ) = 2 ) ) )
149	148	3expib 1268	. . . . 5 |- ( G e. UMGraph -> ( ( { A , B } e. E /\ { B , C } e. E ) -> ( ( A e. V /\ B e. V /\ C e. V ) -> E. f ( f (Walks ` G ) <" A B C "> /\ ( # ` f ) = 2 ) ) ) )
150	149	com23 86	. . . 4 |- ( G e. UMGraph -> ( ( A e. V /\ B e. V /\ C e. V ) -> ( ( { A , B } e. E /\ { B , C } e. E ) -> E. f ( f (Walks ` G ) <" A B C "> /\ ( # ` f ) = 2 ) ) ) )
151	150	imp 445	. . 3 |- ( ( G e. UMGraph /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( ( { A , B } e. E /\ { B , C } e. E ) -> E. f ( f (Walks ` G ) <" A B C "> /\ ( # ` f ) = 2 ) ) )
152	75, 151	impbid 202	. 2 |- ( ( G e. UMGraph /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( E. f ( f (Walks ` G ) <" A B C "> /\ ( # ` f ) = 2 ) <-> ( { A , B } e. E /\ { B , C } e. E ) ) )
153	8, 152	bitrd 268	1 |- ( ( G e. UMGraph /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( <" A B C "> e. ( A ( 2 WWalksNOn G ) C ) <-> ( { A , B } e. E /\ { B , C } e. E ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   {cpr 4179   {ctp 4181   class class class wbr 4653   dom cdm 5114   ran crn 5115   Fun wfun 5882   -->wf 5884   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   2c2 11070   3c3 11071   ...cfz 12326  ..^cfzo 12465   #chash 13117  Word cword 13291   <"cs3 13587  Vtxcvtx 25874  iEdgciedg 25875  Edgcedg 25939   UHGraph cuhgr 25951   UPGraph cupgr 25975   UMGraph cumgr 25976  Walkscwlks 26492   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-umgr 25978  df-wlks 26495  df-wwlks 26722  df-wwlksn 26723  df-wwlksnon 26724

This theorem is referenced by:  wwlks2onsym  26851  usgr2wspthons3  26857  frgr2wwlkeu  27191