Metamath Proof Explorer

Theorem umgrwwlks2on

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)

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

Step Hyp Ref Expression
1 s3wwlks2on.v 
 |-  V = ( Vtx ` G )
2 usgrwwlks2on.e 
 |-  E = ( Edg ` G )
3 umgrupgr 
 |-  ( G e. UMGraph -> G e. UPGraph )
4 3 adantr 
 |-  ( ( G e. UMGraph /\ ( A e. V /\ B e. V /\ C e. V ) ) -> G e. UPGraph )
5 simp1 
 |-  ( ( A e. V /\ B e. V /\ C e. V ) -> A e. V )
6 5 adantl 
 |-  ( ( G e. UMGraph /\ ( A e. V /\ B e. V /\ C e. V ) ) -> A e. V )
7 simpr3 
 |-  ( ( G e. UMGraph /\ ( A e. V /\ B e. V /\ C e. V ) ) -> C e. V )
8 1 s3wwlks2on 
 |-  ( ( 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 ) ) )
9 4 6 7 8 syl3anc 
 |-  ( ( 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 ) ) )
10 eqid 
 |-  ( iEdg ` G ) = ( iEdg ` G )
11 1 10 upgr2wlk 
 |-  ( 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 ) } ) ) ) )
12 3 11 syl 
 |-  ( 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 ) } ) ) ) )
13 12 adantr 
 |-  ( ( 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 ) } ) ) ) )
14 s3fv0 
 |-  ( A e. V -> ( <" A B C "> ` 0 ) = A )
15 14 3ad2ant1 
 |-  ( ( A e. V /\ B e. V /\ C e. V ) -> ( <" A B C "> ` 0 ) = A )
16 s3fv1 
 |-  ( B e. V -> ( <" A B C "> ` 1 ) = B )
17 16 3ad2ant2 
 |-  ( ( A e. V /\ B e. V /\ C e. V ) -> ( <" A B C "> ` 1 ) = B )
18 15 17 preq12d 
 |-  ( ( A e. V /\ B e. V /\ C e. V ) -> { ( <" A B C "> ` 0 ) , ( <" A B C "> ` 1 ) } = { A , B } )
19 18 eqeq2d 
 |-  ( ( 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 } ) )
20 s3fv2 
 |-  ( C e. V -> ( <" A B C "> ` 2 ) = C )
21 20 3ad2ant3 
 |-  ( ( A e. V /\ B e. V /\ C e. V ) -> ( <" A B C "> ` 2 ) = C )
22 17 21 preq12d 
 |-  ( ( A e. V /\ B e. V /\ C e. V ) -> { ( <" A B C "> ` 1 ) , ( <" A B C "> ` 2 ) } = { B , C } )
23 22 eqeq2d 
 |-  ( ( 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 } ) )
24 19 23 anbi12d 
 |-  ( ( 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 adantl 
 |-  ( ( 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 } ) ) )
26 25 3anbi3d 
 |-  ( ( 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 } ) ) ) )
27 umgruhgr 
 |-  ( G e. UMGraph -> G e. UHGraph )
28 10 uhgrfun 
 |-  ( G e. UHGraph -> Fun ( iEdg ` G ) )
29 fdmrn 
 |-  ( Fun ( iEdg ` G ) <-> ( iEdg ` G ) : dom ( iEdg ` G ) --> ran ( iEdg ` G ) )
30 simpr 
 |-  ( ( f : ( 0 ..^ 2 ) --> dom ( iEdg ` G ) /\ ( iEdg ` G ) : dom ( iEdg ` G ) --> ran ( iEdg ` G ) ) -> ( iEdg ` G ) : dom ( iEdg ` G ) --> ran ( iEdg ` G ) )
31 id 
 |-  ( f : ( 0 ..^ 2 ) --> dom ( iEdg ` G ) -> f : ( 0 ..^ 2 ) --> dom ( iEdg ` G ) )
32 0elpr01 
 |-  0 e. { 0 , 1 }
33 fzo0to2pr 
 |-  ( 0 ..^ 2 ) = { 0 , 1 }
34 32 33 eleqtrri 
 |-  0 e. ( 0 ..^ 2 )
35 34 a1i 
 |-  ( f : ( 0 ..^ 2 ) --> dom ( iEdg ` G ) -> 0 e. ( 0 ..^ 2 ) )
36 31 35 ffvelcdmd 
 |-  ( f : ( 0 ..^ 2 ) --> dom ( iEdg ` G ) -> ( f ` 0 ) e. dom ( iEdg ` G ) )
37 36 adantr 
 |-  ( ( f : ( 0 ..^ 2 ) --> dom ( iEdg ` G ) /\ ( iEdg ` G ) : dom ( iEdg ` G ) --> ran ( iEdg ` G ) ) -> ( f ` 0 ) e. dom ( iEdg ` G ) )
38 30 37 ffvelcdmd 
 |-  ( ( f : ( 0 ..^ 2 ) --> dom ( iEdg ` G ) /\ ( iEdg ` G ) : dom ( iEdg ` G ) --> ran ( iEdg ` G ) ) -> ( ( iEdg ` G ) ` ( f ` 0 ) ) e. ran ( iEdg ` G ) )
39 1elpr01 
 |-  1 e. { 0 , 1 }
40 39 33 eleqtrri 
 |-  1 e. ( 0 ..^ 2 )
41 40 a1i 
 |-  ( f : ( 0 ..^ 2 ) --> dom ( iEdg ` G ) -> 1 e. ( 0 ..^ 2 ) )
42 31 41 ffvelcdmd 
 |-  ( f : ( 0 ..^ 2 ) --> dom ( iEdg ` G ) -> ( f ` 1 ) e. dom ( iEdg ` G ) )
43 42 adantr 
 |-  ( ( f : ( 0 ..^ 2 ) --> dom ( iEdg ` G ) /\ ( iEdg ` G ) : dom ( iEdg ` G ) --> ran ( iEdg ` G ) ) -> ( f ` 1 ) e. dom ( iEdg ` G ) )
44 30 43 ffvelcdmd 
 |-  ( ( f : ( 0 ..^ 2 ) --> dom ( iEdg ` G ) /\ ( iEdg ` G ) : dom ( iEdg ` G ) --> ran ( iEdg ` G ) ) -> ( ( iEdg ` G ) ` ( f ` 1 ) ) e. ran ( iEdg ` G ) )
45 38 44 jca 
 |-  ( ( 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 ) ) )
46 45 ex 
 |-  ( 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 3ad2ant1 
 |-  ( ( 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 ) ) ) )
48 47 com12 
 |-  ( ( 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 ) ) ) )
49 29 48 sylbi 
 |-  ( 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 ) ) ) )
50 27 28 49 3syl 
 |-  ( 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 ) ) ) )
51 50 imp 
 |-  ( ( 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 eqcom 
 |-  ( ( ( iEdg ` G ) ` ( f ` 0 ) ) = { A , B } <-> { A , B } = ( ( iEdg ` G ) ` ( f ` 0 ) ) )
53 52 birani 
 |-  ( ( ( ( iEdg ` G ) ` ( f ` 0 ) ) = { A , B } /\ ( ( iEdg ` G ) ` ( f ` 1 ) ) = { B , C } ) -> { A , B } = ( ( iEdg ` G ) ` ( f ` 0 ) ) )
54 53 3ad2ant3 
 |-  ( ( 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 ) ) )
55 54 adantl 
 |-  ( ( 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 ) ) )
56 edgval 
 |-  ( Edg ` G ) = ran ( iEdg ` G )
57 2 56 eqtri 
 |-  E = ran ( iEdg ` G )
58 57 a1i 
 |-  ( ( 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 ) )
59 55 58 eleq12d 
 |-  ( ( 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 ) ) )
60 eqcom 
 |-  ( ( ( iEdg ` G ) ` ( f ` 1 ) ) = { B , C } <-> { B , C } = ( ( iEdg ` G ) ` ( f ` 1 ) ) )
61 60 bilani 
 |-  ( ( ( ( iEdg ` G ) ` ( f ` 0 ) ) = { A , B } /\ ( ( iEdg ` G ) ` ( f ` 1 ) ) = { B , C } ) -> { B , C } = ( ( iEdg ` G ) ` ( f ` 1 ) ) )
62 61 3ad2ant3 
 |-  ( ( 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 ) ) )
63 62 adantl 
 |-  ( ( 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 ) ) )
64 63 58 eleq12d 
 |-  ( ( 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 ) ) )
65 59 64 anbi12d 
 |-  ( ( 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 ) ) ) )
66 51 65 mpbird 
 |-  ( ( 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 ) )
67 66 ex 
 |-  ( 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 ) ) )
68 67 adantr 
 |-  ( ( 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 ) ) )
69 26 68 sylbid 
 |-  ( ( 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 ) ) )
70 13 69 sylbid 
 |-  ( ( 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 ) ) )
71 70 exlimdv 
 |-  ( ( 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 ) ) )
72 2 umgr2wlk 
 |-  ( ( 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 ) ) ) )
73 wlklenvp1 
 |-  ( f ( Walks ` G ) p -> ( # ` p ) = ( ( # ` f ) + 1 ) )
74 oveq1 
 |-  ( ( # ` f ) = 2 -> ( ( # ` f ) + 1 ) = ( 2 + 1 ) )
75 2p1e3 
 |-  ( 2 + 1 ) = 3
76 74 75 eqtrdi 
 |-  ( ( # ` f ) = 2 -> ( ( # ` f ) + 1 ) = 3 )
77 76 adantr 
 |-  ( ( ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) -> ( ( # ` f ) + 1 ) = 3 )
78 73 77 sylan9eq 
 |-  ( ( f ( Walks ` G ) p /\ ( ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) -> ( # ` p ) = 3 )
79 eqcom 
 |-  ( A = ( p ` 0 ) <-> ( p ` 0 ) = A )
80 eqcom 
 |-  ( B = ( p ` 1 ) <-> ( p ` 1 ) = B )
81 eqcom 
 |-  ( C = ( p ` 2 ) <-> ( p ` 2 ) = C )
82 79 80 81 3anbi123i 
 |-  ( ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) <-> ( ( p ` 0 ) = A /\ ( p ` 1 ) = B /\ ( p ` 2 ) = C ) )
83 82 bilani 
 |-  ( ( ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) -> ( ( p ` 0 ) = A /\ ( p ` 1 ) = B /\ ( p ` 2 ) = C ) )
84 83 adantl 
 |-  ( ( f ( Walks ` G ) p /\ ( ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) -> ( ( p ` 0 ) = A /\ ( p ` 1 ) = B /\ ( p ` 2 ) = C ) )
85 78 84 jca 
 |-  ( ( 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 ) ) )
86 1 wlkpwrd 
 |-  ( f ( Walks ` G ) p -> p e. Word V )
87 76 eqeq2d 
 |-  ( ( # ` f ) = 2 -> ( ( # ` p ) = ( ( # ` f ) + 1 ) <-> ( # ` p ) = 3 ) )
88 87 adantl 
 |-  ( ( p e. Word V /\ ( # ` f ) = 2 ) -> ( ( # ` p ) = ( ( # ` f ) + 1 ) <-> ( # ` p ) = 3 ) )
89 simp1 
 |-  ( ( p e. Word V /\ ( # ` p ) = 3 /\ ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) -> p e. Word V )
90 oveq2 
 |-  ( ( # ` p ) = 3 -> ( 0 ..^ ( # ` p ) ) = ( 0 ..^ 3 ) )
91 fzo0to3tp 
 |-  ( 0 ..^ 3 ) = { 0 , 1 , 2 }
92 90 91 eqtrdi 
 |-  ( ( # ` p ) = 3 -> ( 0 ..^ ( # ` p ) ) = { 0 , 1 , 2 } )
93 c0ex 
 |-  0 e. _V
94 93 tpid1 
 |-  0 e. { 0 , 1 , 2 }
95 eleq2 
 |-  ( ( 0 ..^ ( # ` p ) ) = { 0 , 1 , 2 } -> ( 0 e. ( 0 ..^ ( # ` p ) ) <-> 0 e. { 0 , 1 , 2 } ) )
96 94 95 mpbiri 
 |-  ( ( 0 ..^ ( # ` p ) ) = { 0 , 1 , 2 } -> 0 e. ( 0 ..^ ( # ` p ) ) )
97 wrdsymbcl 
 |-  ( ( p e. Word V /\ 0 e. ( 0 ..^ ( # ` p ) ) ) -> ( p ` 0 ) e. V )
98 96 97 sylan2 
 |-  ( ( p e. Word V /\ ( 0 ..^ ( # ` p ) ) = { 0 , 1 , 2 } ) -> ( p ` 0 ) e. V )
99 1ex 
 |-  1 e. _V
100 99 tpid2 
 |-  1 e. { 0 , 1 , 2 }
101 eleq2 
 |-  ( ( 0 ..^ ( # ` p ) ) = { 0 , 1 , 2 } -> ( 1 e. ( 0 ..^ ( # ` p ) ) <-> 1 e. { 0 , 1 , 2 } ) )
102 100 101 mpbiri 
 |-  ( ( 0 ..^ ( # ` p ) ) = { 0 , 1 , 2 } -> 1 e. ( 0 ..^ ( # ` p ) ) )
103 wrdsymbcl 
 |-  ( ( p e. Word V /\ 1 e. ( 0 ..^ ( # ` p ) ) ) -> ( p ` 1 ) e. V )
104 102 103 sylan2 
 |-  ( ( p e. Word V /\ ( 0 ..^ ( # ` p ) ) = { 0 , 1 , 2 } ) -> ( p ` 1 ) e. V )
105 2ex 
 |-  2 e. _V
106 105 tpid3 
 |-  2 e. { 0 , 1 , 2 }
107 eleq2 
 |-  ( ( 0 ..^ ( # ` p ) ) = { 0 , 1 , 2 } -> ( 2 e. ( 0 ..^ ( # ` p ) ) <-> 2 e. { 0 , 1 , 2 } ) )
108 106 107 mpbiri 
 |-  ( ( 0 ..^ ( # ` p ) ) = { 0 , 1 , 2 } -> 2 e. ( 0 ..^ ( # ` p ) ) )
109 wrdsymbcl 
 |-  ( ( p e. Word V /\ 2 e. ( 0 ..^ ( # ` p ) ) ) -> ( p ` 2 ) e. V )
110 108 109 sylan2 
 |-  ( ( p e. Word V /\ ( 0 ..^ ( # ` p ) ) = { 0 , 1 , 2 } ) -> ( p ` 2 ) e. V )
111 98 104 110 3jca 
 |-  ( ( p e. Word V /\ ( 0 ..^ ( # ` p ) ) = { 0 , 1 , 2 } ) -> ( ( p ` 0 ) e. V /\ ( p ` 1 ) e. V /\ ( p ` 2 ) e. V ) )
112 92 111 sylan2 
 |-  ( ( p e. Word V /\ ( # ` p ) = 3 ) -> ( ( p ` 0 ) e. V /\ ( p ` 1 ) e. V /\ ( p ` 2 ) e. V ) )
113 112 3adant3 
 |-  ( ( 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 ) )
114 eleq1 
 |-  ( A = ( p ` 0 ) -> ( A e. V <-> ( p ` 0 ) e. V ) )
115 114 3ad2ant1 
 |-  ( ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) -> ( A e. V <-> ( p ` 0 ) e. V ) )
116 eleq1 
 |-  ( B = ( p ` 1 ) -> ( B e. V <-> ( p ` 1 ) e. V ) )
117 116 3ad2ant2 
 |-  ( ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) -> ( B e. V <-> ( p ` 1 ) e. V ) )
118 eleq1 
 |-  ( C = ( p ` 2 ) -> ( C e. V <-> ( p ` 2 ) e. V ) )
119 118 3ad2ant3 
 |-  ( ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) -> ( C e. V <-> ( p ` 2 ) e. V ) )
120 115 117 119 3anbi123d 
 |-  ( ( 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 ) ) )
121 120 3ad2ant3 
 |-  ( ( 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 ) ) )
122 113 121 mpbird 
 |-  ( ( p e. Word V /\ ( # ` p ) = 3 /\ ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) -> ( A e. V /\ B e. V /\ C e. V ) )
123 89 122 jca 
 |-  ( ( 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 ) ) )
124 123 3exp 
 |-  ( 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 ) ) ) ) )
125 124 adantr 
 |-  ( ( 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 ) ) ) ) )
126 88 125 sylbid 
 |-  ( ( 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 ) ) ) ) )
127 126 impancom 
 |-  ( ( 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 ) ) ) ) )
128 127 impd 
 |-  ( ( 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 ) ) ) )
129 86 73 128 syl2anc 
 |-  ( 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 ) ) ) )
130 129 imp 
 |-  ( ( 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 ) ) )
131 eqwrds3 
 |-  ( ( 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 ) ) ) )
132 130 131 syl 
 |-  ( ( 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 ) ) ) )
133 85 132 mpbird 
 |-  ( ( f ( Walks ` G ) p /\ ( ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) ) -> p = <" A B C "> )
134 133 breq2d 
 |-  ( ( 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 "> ) )
135 134 biimpd 
 |-  ( ( 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 "> ) )
136 135 ex 
 |-  ( 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 "> ) ) )
137 136 pm2.43a 
 |-  ( f ( Walks ` G ) p -> ( ( ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) -> f ( Walks ` G ) <" A B C "> ) )
138 137 3impib 
 |-  ( ( f ( Walks ` G ) p /\ ( # ` f ) = 2 /\ ( A = ( p ` 0 ) /\ B = ( p ` 1 ) /\ C = ( p ` 2 ) ) ) -> f ( Walks ` G ) <" A B C "> )
139 138 adantl 
 |-  ( ( ( 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 "> )
140 simpr2 
 |-  ( ( ( 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 )
141 139 140 jca 
 |-  ( ( ( 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 ) )
142 141 ex 
 |-  ( ( 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 ) ) )
143 142 exlimdv 
 |-  ( ( 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 ) ) )
144 143 eximdv 
 |-  ( ( 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 ) ) )
145 72 144 syl5com 
 |-  ( ( 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 ) ) )
146 145 3expib 
 |-  ( 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 ) ) ) )
147 146 com23 
 |-  ( 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 ) ) ) )
148 147 imp 
 |-  ( ( 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 ) ) )
149 71 148 impbid 
 |-  ( ( 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 ) ) )
150 9 149 bitrd 
 |-  ( ( 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 ) ) )