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 ∈ UMGraph ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) ) → ( ⟨“ 𝐴 𝐵 𝐶 ”⟩ ∈ ( 𝐴 ( 2 WWalksNOn 𝐺 ) 𝐶 ) ↔ ( { 𝐴 , 𝐵 } ∈ 𝐸 ∧ { 𝐵 , 𝐶 } ∈ 𝐸 ) ) )

Proof

Step Hyp Ref Expression
1 s3wwlks2on.v ⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2 usgrwwlks2on.e ⊢ 𝐸 = ( Edg ‘ 𝐺 )
3 umgrupgr ⊢ ( 𝐺 ∈ UMGraph → 𝐺 ∈ UPGraph )
4 3 adantr ⊢ ( ( 𝐺 ∈ UMGraph ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) ) → 𝐺 ∈ UPGraph )
5 simp1 ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) → 𝐴 ∈ 𝑉 )
6 5 adantl ⊢ ( ( 𝐺 ∈ UMGraph ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) ) → 𝐴 ∈ 𝑉 )
7 simpr3 ⊢ ( ( 𝐺 ∈ UMGraph ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) ) → 𝐶 ∈ 𝑉 )
8 1 s3wwlks2on ⊢ ( ( 𝐺 ∈ UPGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) → ( ⟨“ 𝐴 𝐵 𝐶 ”⟩ ∈ ( 𝐴 ( 2 WWalksNOn 𝐺 ) 𝐶 ) ↔ ∃ 𝑓 ( 𝑓 ( Walks ‘ 𝐺 ) ⟨“ 𝐴 𝐵 𝐶 ”⟩ ∧ ( ♯ ‘ 𝑓 ) = 2 ) ) )
9 4 6 7 8 syl3anc ⊢ ( ( 𝐺 ∈ UMGraph ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) ) → ( ⟨“ 𝐴 𝐵 𝐶 ”⟩ ∈ ( 𝐴 ( 2 WWalksNOn 𝐺 ) 𝐶 ) ↔ ∃ 𝑓 ( 𝑓 ( Walks ‘ 𝐺 ) ⟨“ 𝐴 𝐵 𝐶 ”⟩ ∧ ( ♯ ‘ 𝑓 ) = 2 ) ) )
10 eqid ⊢ ( iEdg ‘ 𝐺 ) = ( iEdg ‘ 𝐺 )
11 1 10 upgr2wlk ⊢ ( 𝐺 ∈ 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 ∈ 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 ∈ UMGraph ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 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 ∈ 𝑉 → ( ⟨“ 𝐴 𝐵 𝐶 ”⟩ ‘ 0 ) = 𝐴 )
15 14 3ad2ant1 ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) → ( ⟨“ 𝐴 𝐵 𝐶 ”⟩ ‘ 0 ) = 𝐴 )
16 s3fv1 ⊢ ( 𝐵 ∈ 𝑉 → ( ⟨“ 𝐴 𝐵 𝐶 ”⟩ ‘ 1 ) = 𝐵 )
17 16 3ad2ant2 ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) → ( ⟨“ 𝐴 𝐵 𝐶 ”⟩ ‘ 1 ) = 𝐵 )
18 15 17 preq12d ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) → { ( ⟨“ 𝐴 𝐵 𝐶 ”⟩ ‘ 0 ) , ( ⟨“ 𝐴 𝐵 𝐶 ”⟩ ‘ 1 ) } = { 𝐴 , 𝐵 } )
19 18 eqeq2d ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) → ( ( ( iEdg ‘ 𝐺 ) ‘ ( 𝑓 ‘ 0 ) ) = { ( ⟨“ 𝐴 𝐵 𝐶 ”⟩ ‘ 0 ) , ( ⟨“ 𝐴 𝐵 𝐶 ”⟩ ‘ 1 ) } ↔ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝑓 ‘ 0 ) ) = { 𝐴 , 𝐵 } ) )
20 s3fv2 ⊢ ( 𝐶 ∈ 𝑉 → ( ⟨“ 𝐴 𝐵 𝐶 ”⟩ ‘ 2 ) = 𝐶 )
21 20 3ad2ant3 ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) → ( ⟨“ 𝐴 𝐵 𝐶 ”⟩ ‘ 2 ) = 𝐶 )
22 17 21 preq12d ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) → { ( ⟨“ 𝐴 𝐵 𝐶 ”⟩ ‘ 1 ) , ( ⟨“ 𝐴 𝐵 𝐶 ”⟩ ‘ 2 ) } = { 𝐵 , 𝐶 } )
23 22 eqeq2d ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) → ( ( ( iEdg ‘ 𝐺 ) ‘ ( 𝑓 ‘ 1 ) ) = { ( ⟨“ 𝐴 𝐵 𝐶 ”⟩ ‘ 1 ) , ( ⟨“ 𝐴 𝐵 𝐶 ”⟩ ‘ 2 ) } ↔ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝑓 ‘ 1 ) ) = { 𝐵 , 𝐶 } ) )
24 19 23 anbi12d ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 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 ∈ UMGraph ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 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 ∈ UMGraph ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 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 ∈ UMGraph → 𝐺 ∈ UHGraph )
28 10 uhgrfun ⊢ ( 𝐺 ∈ 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 c0ex ⊢ 0 ∈ V
33 32 prid1 ⊢ 0 ∈ { 0 , 1 }
34 fzo0to2pr ⊢ ( 0 ..^ 2 ) = { 0 , 1 }
35 33 34 eleqtrri ⊢ 0 ∈ ( 0 ..^ 2 )
36 35 a1i ⊢ ( 𝑓 : ( 0 ..^ 2 ) ⟶ dom ( iEdg ‘ 𝐺 ) → 0 ∈ ( 0 ..^ 2 ) )
37 31 36 ffvelrnd ⊢ ( 𝑓 : ( 0 ..^ 2 ) ⟶ dom ( iEdg ‘ 𝐺 ) → ( 𝑓 ‘ 0 ) ∈ dom ( iEdg ‘ 𝐺 ) )
38 37 adantr ⊢ ( ( 𝑓 : ( 0 ..^ 2 ) ⟶ dom ( iEdg ‘ 𝐺 ) ∧ ( iEdg ‘ 𝐺 ) : dom ( iEdg ‘ 𝐺 ) ⟶ ran ( iEdg ‘ 𝐺 ) ) → ( 𝑓 ‘ 0 ) ∈ dom ( iEdg ‘ 𝐺 ) )
39 30 38 ffvelrnd ⊢ ( ( 𝑓 : ( 0 ..^ 2 ) ⟶ dom ( iEdg ‘ 𝐺 ) ∧ ( iEdg ‘ 𝐺 ) : dom ( iEdg ‘ 𝐺 ) ⟶ ran ( iEdg ‘ 𝐺 ) ) → ( ( iEdg ‘ 𝐺 ) ‘ ( 𝑓 ‘ 0 ) ) ∈ ran ( iEdg ‘ 𝐺 ) )
40 1ex ⊢ 1 ∈ V
41 40 prid2 ⊢ 1 ∈ { 0 , 1 }
42 41 34 eleqtrri ⊢ 1 ∈ ( 0 ..^ 2 )
43 42 a1i ⊢ ( 𝑓 : ( 0 ..^ 2 ) ⟶ dom ( iEdg ‘ 𝐺 ) → 1 ∈ ( 0 ..^ 2 ) )
44 31 43 ffvelrnd ⊢ ( 𝑓 : ( 0 ..^ 2 ) ⟶ dom ( iEdg ‘ 𝐺 ) → ( 𝑓 ‘ 1 ) ∈ dom ( iEdg ‘ 𝐺 ) )
45 44 adantr ⊢ ( ( 𝑓 : ( 0 ..^ 2 ) ⟶ dom ( iEdg ‘ 𝐺 ) ∧ ( iEdg ‘ 𝐺 ) : dom ( iEdg ‘ 𝐺 ) ⟶ ran ( iEdg ‘ 𝐺 ) ) → ( 𝑓 ‘ 1 ) ∈ dom ( iEdg ‘ 𝐺 ) )
46 30 45 ffvelrnd ⊢ ( ( 𝑓 : ( 0 ..^ 2 ) ⟶ dom ( iEdg ‘ 𝐺 ) ∧ ( iEdg ‘ 𝐺 ) : dom ( iEdg ‘ 𝐺 ) ⟶ ran ( iEdg ‘ 𝐺 ) ) → ( ( iEdg ‘ 𝐺 ) ‘ ( 𝑓 ‘ 1 ) ) ∈ ran ( iEdg ‘ 𝐺 ) )
47 39 46 jca ⊢ ( ( 𝑓 : ( 0 ..^ 2 ) ⟶ dom ( iEdg ‘ 𝐺 ) ∧ ( iEdg ‘ 𝐺 ) : dom ( iEdg ‘ 𝐺 ) ⟶ ran ( iEdg ‘ 𝐺 ) ) → ( ( ( iEdg ‘ 𝐺 ) ‘ ( 𝑓 ‘ 0 ) ) ∈ ran ( iEdg ‘ 𝐺 ) ∧ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝑓 ‘ 1 ) ) ∈ ran ( iEdg ‘ 𝐺 ) ) )
48 47 ex ⊢ ( 𝑓 : ( 0 ..^ 2 ) ⟶ dom ( iEdg ‘ 𝐺 ) → ( ( iEdg ‘ 𝐺 ) : dom ( iEdg ‘ 𝐺 ) ⟶ ran ( iEdg ‘ 𝐺 ) → ( ( ( iEdg ‘ 𝐺 ) ‘ ( 𝑓 ‘ 0 ) ) ∈ ran ( iEdg ‘ 𝐺 ) ∧ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝑓 ‘ 1 ) ) ∈ ran ( iEdg ‘ 𝐺 ) ) ) )
49 48 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 ) ) ∈ ran ( iEdg ‘ 𝐺 ) ∧ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝑓 ‘ 1 ) ) ∈ ran ( iEdg ‘ 𝐺 ) ) ) )
50 49 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 ) ) ∈ ran ( iEdg ‘ 𝐺 ) ∧ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝑓 ‘ 1 ) ) ∈ ran ( iEdg ‘ 𝐺 ) ) ) )
51 29 50 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 ) ) ∈ ran ( iEdg ‘ 𝐺 ) ∧ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝑓 ‘ 1 ) ) ∈ ran ( iEdg ‘ 𝐺 ) ) ) )
52 27 28 51 3syl ⊢ ( 𝐺 ∈ 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 ) ) ∈ ran ( iEdg ‘ 𝐺 ) ∧ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝑓 ‘ 1 ) ) ∈ ran ( iEdg ‘ 𝐺 ) ) ) )
53 52 imp ⊢ ( ( 𝐺 ∈ 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 ) ) ∈ ran ( iEdg ‘ 𝐺 ) ∧ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝑓 ‘ 1 ) ) ∈ ran ( iEdg ‘ 𝐺 ) ) )
54 eqcom ⊢ ( ( ( iEdg ‘ 𝐺 ) ‘ ( 𝑓 ‘ 0 ) ) = { 𝐴 , 𝐵 } ↔ { 𝐴 , 𝐵 } = ( ( iEdg ‘ 𝐺 ) ‘ ( 𝑓 ‘ 0 ) ) )
55 54 biimpi ⊢ ( ( ( iEdg ‘ 𝐺 ) ‘ ( 𝑓 ‘ 0 ) ) = { 𝐴 , 𝐵 } → { 𝐴 , 𝐵 } = ( ( iEdg ‘ 𝐺 ) ‘ ( 𝑓 ‘ 0 ) ) )
56 55 adantr ⊢ ( ( ( ( iEdg ‘ 𝐺 ) ‘ ( 𝑓 ‘ 0 ) ) = { 𝐴 , 𝐵 } ∧ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝑓 ‘ 1 ) ) = { 𝐵 , 𝐶 } ) → { 𝐴 , 𝐵 } = ( ( iEdg ‘ 𝐺 ) ‘ ( 𝑓 ‘ 0 ) ) )
57 56 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 ) ) )
58 57 adantl ⊢ ( ( 𝐺 ∈ 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 ) ) )
59 edgval ⊢ ( Edg ‘ 𝐺 ) = ran ( iEdg ‘ 𝐺 )
60 2 59 eqtri ⊢ 𝐸 = ran ( iEdg ‘ 𝐺 )
61 60 a1i ⊢ ( ( 𝐺 ∈ 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 58 61 eleq12d ⊢ ( ( 𝐺 ∈ 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 ↔ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝑓 ‘ 0 ) ) ∈ ran ( iEdg ‘ 𝐺 ) ) )
63 eqcom ⊢ ( ( ( iEdg ‘ 𝐺 ) ‘ ( 𝑓 ‘ 1 ) ) = { 𝐵 , 𝐶 } ↔ { 𝐵 , 𝐶 } = ( ( iEdg ‘ 𝐺 ) ‘ ( 𝑓 ‘ 1 ) ) )
64 63 biimpi ⊢ ( ( ( iEdg ‘ 𝐺 ) ‘ ( 𝑓 ‘ 1 ) ) = { 𝐵 , 𝐶 } → { 𝐵 , 𝐶 } = ( ( iEdg ‘ 𝐺 ) ‘ ( 𝑓 ‘ 1 ) ) )
65 64 adantl ⊢ ( ( ( ( iEdg ‘ 𝐺 ) ‘ ( 𝑓 ‘ 0 ) ) = { 𝐴 , 𝐵 } ∧ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝑓 ‘ 1 ) ) = { 𝐵 , 𝐶 } ) → { 𝐵 , 𝐶 } = ( ( iEdg ‘ 𝐺 ) ‘ ( 𝑓 ‘ 1 ) ) )
66 65 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 ) ) )
67 66 adantl ⊢ ( ( 𝐺 ∈ 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 ⊢ ( ( 𝐺 ∈ 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 ↔ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝑓 ‘ 1 ) ) ∈ ran ( iEdg ‘ 𝐺 ) ) )
69 62 68 anbi12d ⊢ ( ( 𝐺 ∈ 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 ∧ { 𝐵 , 𝐶 } ∈ 𝐸 ) ↔ ( ( ( iEdg ‘ 𝐺 ) ‘ ( 𝑓 ‘ 0 ) ) ∈ ran ( iEdg ‘ 𝐺 ) ∧ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝑓 ‘ 1 ) ) ∈ ran ( iEdg ‘ 𝐺 ) ) ) )
70 53 69 mpbird ⊢ ( ( 𝐺 ∈ 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 ∧ { 𝐵 , 𝐶 } ∈ 𝐸 ) )
71 70 ex ⊢ ( 𝐺 ∈ 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 ∧ { 𝐵 , 𝐶 } ∈ 𝐸 ) ) )
72 71 adantr ⊢ ( ( 𝐺 ∈ UMGraph ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 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 ∧ { 𝐵 , 𝐶 } ∈ 𝐸 ) ) )
73 26 72 sylbid ⊢ ( ( 𝐺 ∈ UMGraph ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 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 ∧ { 𝐵 , 𝐶 } ∈ 𝐸 ) ) )
74 13 73 sylbid ⊢ ( ( 𝐺 ∈ UMGraph ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) ) → ( ( 𝑓 ( Walks ‘ 𝐺 ) ⟨“ 𝐴 𝐵 𝐶 ”⟩ ∧ ( ♯ ‘ 𝑓 ) = 2 ) → ( { 𝐴 , 𝐵 } ∈ 𝐸 ∧ { 𝐵 , 𝐶 } ∈ 𝐸 ) ) )
75 74 exlimdv ⊢ ( ( 𝐺 ∈ UMGraph ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) ) → ( ∃ 𝑓 ( 𝑓 ( Walks ‘ 𝐺 ) ⟨“ 𝐴 𝐵 𝐶 ”⟩ ∧ ( ♯ ‘ 𝑓 ) = 2 ) → ( { 𝐴 , 𝐵 } ∈ 𝐸 ∧ { 𝐵 , 𝐶 } ∈ 𝐸 ) ) )
76 2 umgr2wlk ⊢ ( ( 𝐺 ∈ UMGraph ∧ { 𝐴 , 𝐵 } ∈ 𝐸 ∧ { 𝐵 , 𝐶 } ∈ 𝐸 ) → ∃ 𝑓 ∃ 𝑝 ( 𝑓 ( Walks ‘ 𝐺 ) 𝑝 ∧ ( ♯ ‘ 𝑓 ) = 2 ∧ ( 𝐴 = ( 𝑝 ‘ 0 ) ∧ 𝐵 = ( 𝑝 ‘ 1 ) ∧ 𝐶 = ( 𝑝 ‘ 2 ) ) ) )
77 wlklenvp1 ⊢ ( 𝑓 ( Walks ‘ 𝐺 ) 𝑝 → ( ♯ ‘ 𝑝 ) = ( ( ♯ ‘ 𝑓 ) + 1 ) )
78 oveq1 ⊢ ( ( ♯ ‘ 𝑓 ) = 2 → ( ( ♯ ‘ 𝑓 ) + 1 ) = ( 2 + 1 ) )
79 2p1e3 ⊢ ( 2 + 1 ) = 3
80 78 79 eqtrdi ⊢ ( ( ♯ ‘ 𝑓 ) = 2 → ( ( ♯ ‘ 𝑓 ) + 1 ) = 3 )
81 80 adantr ⊢ ( ( ( ♯ ‘ 𝑓 ) = 2 ∧ ( 𝐴 = ( 𝑝 ‘ 0 ) ∧ 𝐵 = ( 𝑝 ‘ 1 ) ∧ 𝐶 = ( 𝑝 ‘ 2 ) ) ) → ( ( ♯ ‘ 𝑓 ) + 1 ) = 3 )
82 77 81 sylan9eq ⊢ ( ( 𝑓 ( Walks ‘ 𝐺 ) 𝑝 ∧ ( ( ♯ ‘ 𝑓 ) = 2 ∧ ( 𝐴 = ( 𝑝 ‘ 0 ) ∧ 𝐵 = ( 𝑝 ‘ 1 ) ∧ 𝐶 = ( 𝑝 ‘ 2 ) ) ) ) → ( ♯ ‘ 𝑝 ) = 3 )
83 eqcom ⊢ ( 𝐴 = ( 𝑝 ‘ 0 ) ↔ ( 𝑝 ‘ 0 ) = 𝐴 )
84 eqcom ⊢ ( 𝐵 = ( 𝑝 ‘ 1 ) ↔ ( 𝑝 ‘ 1 ) = 𝐵 )
85 eqcom ⊢ ( 𝐶 = ( 𝑝 ‘ 2 ) ↔ ( 𝑝 ‘ 2 ) = 𝐶 )
86 83 84 85 3anbi123i ⊢ ( ( 𝐴 = ( 𝑝 ‘ 0 ) ∧ 𝐵 = ( 𝑝 ‘ 1 ) ∧ 𝐶 = ( 𝑝 ‘ 2 ) ) ↔ ( ( 𝑝 ‘ 0 ) = 𝐴 ∧ ( 𝑝 ‘ 1 ) = 𝐵 ∧ ( 𝑝 ‘ 2 ) = 𝐶 ) )
87 86 biimpi ⊢ ( ( 𝐴 = ( 𝑝 ‘ 0 ) ∧ 𝐵 = ( 𝑝 ‘ 1 ) ∧ 𝐶 = ( 𝑝 ‘ 2 ) ) → ( ( 𝑝 ‘ 0 ) = 𝐴 ∧ ( 𝑝 ‘ 1 ) = 𝐵 ∧ ( 𝑝 ‘ 2 ) = 𝐶 ) )
88 87 adantl ⊢ ( ( ( ♯ ‘ 𝑓 ) = 2 ∧ ( 𝐴 = ( 𝑝 ‘ 0 ) ∧ 𝐵 = ( 𝑝 ‘ 1 ) ∧ 𝐶 = ( 𝑝 ‘ 2 ) ) ) → ( ( 𝑝 ‘ 0 ) = 𝐴 ∧ ( 𝑝 ‘ 1 ) = 𝐵 ∧ ( 𝑝 ‘ 2 ) = 𝐶 ) )
89 88 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 ) )
90 82 89 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 ) ) )
91 1 wlkpwrd ⊢ ( 𝑓 ( Walks ‘ 𝐺 ) 𝑝 → 𝑝 ∈ Word 𝑉 )
92 80 eqeq2d ⊢ ( ( ♯ ‘ 𝑓 ) = 2 → ( ( ♯ ‘ 𝑝 ) = ( ( ♯ ‘ 𝑓 ) + 1 ) ↔ ( ♯ ‘ 𝑝 ) = 3 ) )
93 92 adantl ⊢ ( ( 𝑝 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑓 ) = 2 ) → ( ( ♯ ‘ 𝑝 ) = ( ( ♯ ‘ 𝑓 ) + 1 ) ↔ ( ♯ ‘ 𝑝 ) = 3 ) )
94 simp1 ⊢ ( ( 𝑝 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑝 ) = 3 ∧ ( 𝐴 = ( 𝑝 ‘ 0 ) ∧ 𝐵 = ( 𝑝 ‘ 1 ) ∧ 𝐶 = ( 𝑝 ‘ 2 ) ) ) → 𝑝 ∈ Word 𝑉 )
95 oveq2 ⊢ ( ( ♯ ‘ 𝑝 ) = 3 → ( 0 ..^ ( ♯ ‘ 𝑝 ) ) = ( 0 ..^ 3 ) )
96 fzo0to3tp ⊢ ( 0 ..^ 3 ) = { 0 , 1 , 2 }
97 95 96 eqtrdi ⊢ ( ( ♯ ‘ 𝑝 ) = 3 → ( 0 ..^ ( ♯ ‘ 𝑝 ) ) = { 0 , 1 , 2 } )
98 32 tpid1 ⊢ 0 ∈ { 0 , 1 , 2 }
99 eleq2 ⊢ ( ( 0 ..^ ( ♯ ‘ 𝑝 ) ) = { 0 , 1 , 2 } → ( 0 ∈ ( 0 ..^ ( ♯ ‘ 𝑝 ) ) ↔ 0 ∈ { 0 , 1 , 2 } ) )
100 98 99 mpbiri ⊢ ( ( 0 ..^ ( ♯ ‘ 𝑝 ) ) = { 0 , 1 , 2 } → 0 ∈ ( 0 ..^ ( ♯ ‘ 𝑝 ) ) )
101 wrdsymbcl ⊢ ( ( 𝑝 ∈ Word 𝑉 ∧ 0 ∈ ( 0 ..^ ( ♯ ‘ 𝑝 ) ) ) → ( 𝑝 ‘ 0 ) ∈ 𝑉 )
102 100 101 sylan2 ⊢ ( ( 𝑝 ∈ Word 𝑉 ∧ ( 0 ..^ ( ♯ ‘ 𝑝 ) ) = { 0 , 1 , 2 } ) → ( 𝑝 ‘ 0 ) ∈ 𝑉 )
103 40 tpid2 ⊢ 1 ∈ { 0 , 1 , 2 }
104 eleq2 ⊢ ( ( 0 ..^ ( ♯ ‘ 𝑝 ) ) = { 0 , 1 , 2 } → ( 1 ∈ ( 0 ..^ ( ♯ ‘ 𝑝 ) ) ↔ 1 ∈ { 0 , 1 , 2 } ) )
105 103 104 mpbiri ⊢ ( ( 0 ..^ ( ♯ ‘ 𝑝 ) ) = { 0 , 1 , 2 } → 1 ∈ ( 0 ..^ ( ♯ ‘ 𝑝 ) ) )
106 wrdsymbcl ⊢ ( ( 𝑝 ∈ Word 𝑉 ∧ 1 ∈ ( 0 ..^ ( ♯ ‘ 𝑝 ) ) ) → ( 𝑝 ‘ 1 ) ∈ 𝑉 )
107 105 106 sylan2 ⊢ ( ( 𝑝 ∈ Word 𝑉 ∧ ( 0 ..^ ( ♯ ‘ 𝑝 ) ) = { 0 , 1 , 2 } ) → ( 𝑝 ‘ 1 ) ∈ 𝑉 )
108 2ex ⊢ 2 ∈ V
109 108 tpid3 ⊢ 2 ∈ { 0 , 1 , 2 }
110 eleq2 ⊢ ( ( 0 ..^ ( ♯ ‘ 𝑝 ) ) = { 0 , 1 , 2 } → ( 2 ∈ ( 0 ..^ ( ♯ ‘ 𝑝 ) ) ↔ 2 ∈ { 0 , 1 , 2 } ) )
111 109 110 mpbiri ⊢ ( ( 0 ..^ ( ♯ ‘ 𝑝 ) ) = { 0 , 1 , 2 } → 2 ∈ ( 0 ..^ ( ♯ ‘ 𝑝 ) ) )
112 wrdsymbcl ⊢ ( ( 𝑝 ∈ Word 𝑉 ∧ 2 ∈ ( 0 ..^ ( ♯ ‘ 𝑝 ) ) ) → ( 𝑝 ‘ 2 ) ∈ 𝑉 )
113 111 112 sylan2 ⊢ ( ( 𝑝 ∈ Word 𝑉 ∧ ( 0 ..^ ( ♯ ‘ 𝑝 ) ) = { 0 , 1 , 2 } ) → ( 𝑝 ‘ 2 ) ∈ 𝑉 )
114 102 107 113 3jca ⊢ ( ( 𝑝 ∈ Word 𝑉 ∧ ( 0 ..^ ( ♯ ‘ 𝑝 ) ) = { 0 , 1 , 2 } ) → ( ( 𝑝 ‘ 0 ) ∈ 𝑉 ∧ ( 𝑝 ‘ 1 ) ∈ 𝑉 ∧ ( 𝑝 ‘ 2 ) ∈ 𝑉 ) )
115 97 114 sylan2 ⊢ ( ( 𝑝 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑝 ) = 3 ) → ( ( 𝑝 ‘ 0 ) ∈ 𝑉 ∧ ( 𝑝 ‘ 1 ) ∈ 𝑉 ∧ ( 𝑝 ‘ 2 ) ∈ 𝑉 ) )
116 115 3adant3 ⊢ ( ( 𝑝 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑝 ) = 3 ∧ ( 𝐴 = ( 𝑝 ‘ 0 ) ∧ 𝐵 = ( 𝑝 ‘ 1 ) ∧ 𝐶 = ( 𝑝 ‘ 2 ) ) ) → ( ( 𝑝 ‘ 0 ) ∈ 𝑉 ∧ ( 𝑝 ‘ 1 ) ∈ 𝑉 ∧ ( 𝑝 ‘ 2 ) ∈ 𝑉 ) )
117 eleq1 ⊢ ( 𝐴 = ( 𝑝 ‘ 0 ) → ( 𝐴 ∈ 𝑉 ↔ ( 𝑝 ‘ 0 ) ∈ 𝑉 ) )
118 117 3ad2ant1 ⊢ ( ( 𝐴 = ( 𝑝 ‘ 0 ) ∧ 𝐵 = ( 𝑝 ‘ 1 ) ∧ 𝐶 = ( 𝑝 ‘ 2 ) ) → ( 𝐴 ∈ 𝑉 ↔ ( 𝑝 ‘ 0 ) ∈ 𝑉 ) )
119 eleq1 ⊢ ( 𝐵 = ( 𝑝 ‘ 1 ) → ( 𝐵 ∈ 𝑉 ↔ ( 𝑝 ‘ 1 ) ∈ 𝑉 ) )
120 119 3ad2ant2 ⊢ ( ( 𝐴 = ( 𝑝 ‘ 0 ) ∧ 𝐵 = ( 𝑝 ‘ 1 ) ∧ 𝐶 = ( 𝑝 ‘ 2 ) ) → ( 𝐵 ∈ 𝑉 ↔ ( 𝑝 ‘ 1 ) ∈ 𝑉 ) )
121 eleq1 ⊢ ( 𝐶 = ( 𝑝 ‘ 2 ) → ( 𝐶 ∈ 𝑉 ↔ ( 𝑝 ‘ 2 ) ∈ 𝑉 ) )
122 121 3ad2ant3 ⊢ ( ( 𝐴 = ( 𝑝 ‘ 0 ) ∧ 𝐵 = ( 𝑝 ‘ 1 ) ∧ 𝐶 = ( 𝑝 ‘ 2 ) ) → ( 𝐶 ∈ 𝑉 ↔ ( 𝑝 ‘ 2 ) ∈ 𝑉 ) )
123 118 120 122 3anbi123d ⊢ ( ( 𝐴 = ( 𝑝 ‘ 0 ) ∧ 𝐵 = ( 𝑝 ‘ 1 ) ∧ 𝐶 = ( 𝑝 ‘ 2 ) ) → ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) ↔ ( ( 𝑝 ‘ 0 ) ∈ 𝑉 ∧ ( 𝑝 ‘ 1 ) ∈ 𝑉 ∧ ( 𝑝 ‘ 2 ) ∈ 𝑉 ) ) )
124 123 3ad2ant3 ⊢ ( ( 𝑝 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑝 ) = 3 ∧ ( 𝐴 = ( 𝑝 ‘ 0 ) ∧ 𝐵 = ( 𝑝 ‘ 1 ) ∧ 𝐶 = ( 𝑝 ‘ 2 ) ) ) → ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) ↔ ( ( 𝑝 ‘ 0 ) ∈ 𝑉 ∧ ( 𝑝 ‘ 1 ) ∈ 𝑉 ∧ ( 𝑝 ‘ 2 ) ∈ 𝑉 ) ) )
125 116 124 mpbird ⊢ ( ( 𝑝 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑝 ) = 3 ∧ ( 𝐴 = ( 𝑝 ‘ 0 ) ∧ 𝐵 = ( 𝑝 ‘ 1 ) ∧ 𝐶 = ( 𝑝 ‘ 2 ) ) ) → ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) )
126 94 125 jca ⊢ ( ( 𝑝 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑝 ) = 3 ∧ ( 𝐴 = ( 𝑝 ‘ 0 ) ∧ 𝐵 = ( 𝑝 ‘ 1 ) ∧ 𝐶 = ( 𝑝 ‘ 2 ) ) ) → ( 𝑝 ∈ Word 𝑉 ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) ) )
127 126 3exp ⊢ ( 𝑝 ∈ Word 𝑉 → ( ( ♯ ‘ 𝑝 ) = 3 → ( ( 𝐴 = ( 𝑝 ‘ 0 ) ∧ 𝐵 = ( 𝑝 ‘ 1 ) ∧ 𝐶 = ( 𝑝 ‘ 2 ) ) → ( 𝑝 ∈ Word 𝑉 ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) ) ) ) )
128 127 adantr ⊢ ( ( 𝑝 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑓 ) = 2 ) → ( ( ♯ ‘ 𝑝 ) = 3 → ( ( 𝐴 = ( 𝑝 ‘ 0 ) ∧ 𝐵 = ( 𝑝 ‘ 1 ) ∧ 𝐶 = ( 𝑝 ‘ 2 ) ) → ( 𝑝 ∈ Word 𝑉 ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) ) ) ) )
129 93 128 sylbid ⊢ ( ( 𝑝 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑓 ) = 2 ) → ( ( ♯ ‘ 𝑝 ) = ( ( ♯ ‘ 𝑓 ) + 1 ) → ( ( 𝐴 = ( 𝑝 ‘ 0 ) ∧ 𝐵 = ( 𝑝 ‘ 1 ) ∧ 𝐶 = ( 𝑝 ‘ 2 ) ) → ( 𝑝 ∈ Word 𝑉 ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) ) ) ) )
130 129 impancom ⊢ ( ( 𝑝 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑝 ) = ( ( ♯ ‘ 𝑓 ) + 1 ) ) → ( ( ♯ ‘ 𝑓 ) = 2 → ( ( 𝐴 = ( 𝑝 ‘ 0 ) ∧ 𝐵 = ( 𝑝 ‘ 1 ) ∧ 𝐶 = ( 𝑝 ‘ 2 ) ) → ( 𝑝 ∈ Word 𝑉 ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) ) ) ) )
131 130 impd ⊢ ( ( 𝑝 ∈ Word 𝑉 ∧ ( ♯ ‘ 𝑝 ) = ( ( ♯ ‘ 𝑓 ) + 1 ) ) → ( ( ( ♯ ‘ 𝑓 ) = 2 ∧ ( 𝐴 = ( 𝑝 ‘ 0 ) ∧ 𝐵 = ( 𝑝 ‘ 1 ) ∧ 𝐶 = ( 𝑝 ‘ 2 ) ) ) → ( 𝑝 ∈ Word 𝑉 ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) ) ) )
132 91 77 131 syl2anc ⊢ ( 𝑓 ( Walks ‘ 𝐺 ) 𝑝 → ( ( ( ♯ ‘ 𝑓 ) = 2 ∧ ( 𝐴 = ( 𝑝 ‘ 0 ) ∧ 𝐵 = ( 𝑝 ‘ 1 ) ∧ 𝐶 = ( 𝑝 ‘ 2 ) ) ) → ( 𝑝 ∈ Word 𝑉 ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) ) ) )
133 132 imp ⊢ ( ( 𝑓 ( Walks ‘ 𝐺 ) 𝑝 ∧ ( ( ♯ ‘ 𝑓 ) = 2 ∧ ( 𝐴 = ( 𝑝 ‘ 0 ) ∧ 𝐵 = ( 𝑝 ‘ 1 ) ∧ 𝐶 = ( 𝑝 ‘ 2 ) ) ) ) → ( 𝑝 ∈ Word 𝑉 ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) ) )
134 eqwrds3 ⊢ ( ( 𝑝 ∈ Word 𝑉 ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) ) → ( 𝑝 = ⟨“ 𝐴 𝐵 𝐶 ”⟩ ↔ ( ( ♯ ‘ 𝑝 ) = 3 ∧ ( ( 𝑝 ‘ 0 ) = 𝐴 ∧ ( 𝑝 ‘ 1 ) = 𝐵 ∧ ( 𝑝 ‘ 2 ) = 𝐶 ) ) ) )
135 133 134 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 ) ) ) )
136 90 135 mpbird ⊢ ( ( 𝑓 ( Walks ‘ 𝐺 ) 𝑝 ∧ ( ( ♯ ‘ 𝑓 ) = 2 ∧ ( 𝐴 = ( 𝑝 ‘ 0 ) ∧ 𝐵 = ( 𝑝 ‘ 1 ) ∧ 𝐶 = ( 𝑝 ‘ 2 ) ) ) ) → 𝑝 = ⟨“ 𝐴 𝐵 𝐶 ”⟩ )
137 136 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 ”⟩ ) )
138 137 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 ”⟩ ) )
139 138 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 ”⟩ ) ) )
140 139 pm2.43a ⊢ ( 𝑓 ( Walks ‘ 𝐺 ) 𝑝 → ( ( ( ♯ ‘ 𝑓 ) = 2 ∧ ( 𝐴 = ( 𝑝 ‘ 0 ) ∧ 𝐵 = ( 𝑝 ‘ 1 ) ∧ 𝐶 = ( 𝑝 ‘ 2 ) ) ) → 𝑓 ( Walks ‘ 𝐺 ) ⟨“ 𝐴 𝐵 𝐶 ”⟩ ) )
141 140 3impib ⊢ ( ( 𝑓 ( Walks ‘ 𝐺 ) 𝑝 ∧ ( ♯ ‘ 𝑓 ) = 2 ∧ ( 𝐴 = ( 𝑝 ‘ 0 ) ∧ 𝐵 = ( 𝑝 ‘ 1 ) ∧ 𝐶 = ( 𝑝 ‘ 2 ) ) ) → 𝑓 ( Walks ‘ 𝐺 ) ⟨“ 𝐴 𝐵 𝐶 ”⟩ )
142 141 adantl ⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) ∧ ( 𝑓 ( Walks ‘ 𝐺 ) 𝑝 ∧ ( ♯ ‘ 𝑓 ) = 2 ∧ ( 𝐴 = ( 𝑝 ‘ 0 ) ∧ 𝐵 = ( 𝑝 ‘ 1 ) ∧ 𝐶 = ( 𝑝 ‘ 2 ) ) ) ) → 𝑓 ( Walks ‘ 𝐺 ) ⟨“ 𝐴 𝐵 𝐶 ”⟩ )
143 simpr2 ⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) ∧ ( 𝑓 ( Walks ‘ 𝐺 ) 𝑝 ∧ ( ♯ ‘ 𝑓 ) = 2 ∧ ( 𝐴 = ( 𝑝 ‘ 0 ) ∧ 𝐵 = ( 𝑝 ‘ 1 ) ∧ 𝐶 = ( 𝑝 ‘ 2 ) ) ) ) → ( ♯ ‘ 𝑓 ) = 2 )
144 142 143 jca ⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 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 ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 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 ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) → ( ∃ 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 ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) → ( ∃ 𝑓 ∃ 𝑝 ( 𝑓 ( Walks ‘ 𝐺 ) 𝑝 ∧ ( ♯ ‘ 𝑓 ) = 2 ∧ ( 𝐴 = ( 𝑝 ‘ 0 ) ∧ 𝐵 = ( 𝑝 ‘ 1 ) ∧ 𝐶 = ( 𝑝 ‘ 2 ) ) ) → ∃ 𝑓 ( 𝑓 ( Walks ‘ 𝐺 ) ⟨“ 𝐴 𝐵 𝐶 ”⟩ ∧ ( ♯ ‘ 𝑓 ) = 2 ) ) )
148 76 147 syl5com ⊢ ( ( 𝐺 ∈ UMGraph ∧ { 𝐴 , 𝐵 } ∈ 𝐸 ∧ { 𝐵 , 𝐶 } ∈ 𝐸 ) → ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) → ∃ 𝑓 ( 𝑓 ( Walks ‘ 𝐺 ) ⟨“ 𝐴 𝐵 𝐶 ”⟩ ∧ ( ♯ ‘ 𝑓 ) = 2 ) ) )
149 148 3expib ⊢ ( 𝐺 ∈ UMGraph → ( ( { 𝐴 , 𝐵 } ∈ 𝐸 ∧ { 𝐵 , 𝐶 } ∈ 𝐸 ) → ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) → ∃ 𝑓 ( 𝑓 ( Walks ‘ 𝐺 ) ⟨“ 𝐴 𝐵 𝐶 ”⟩ ∧ ( ♯ ‘ 𝑓 ) = 2 ) ) ) )
150 149 com23 ⊢ ( 𝐺 ∈ UMGraph → ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) → ( ( { 𝐴 , 𝐵 } ∈ 𝐸 ∧ { 𝐵 , 𝐶 } ∈ 𝐸 ) → ∃ 𝑓 ( 𝑓 ( Walks ‘ 𝐺 ) ⟨“ 𝐴 𝐵 𝐶 ”⟩ ∧ ( ♯ ‘ 𝑓 ) = 2 ) ) ) )
151 150 imp ⊢ ( ( 𝐺 ∈ UMGraph ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) ) → ( ( { 𝐴 , 𝐵 } ∈ 𝐸 ∧ { 𝐵 , 𝐶 } ∈ 𝐸 ) → ∃ 𝑓 ( 𝑓 ( Walks ‘ 𝐺 ) ⟨“ 𝐴 𝐵 𝐶 ”⟩ ∧ ( ♯ ‘ 𝑓 ) = 2 ) ) )
152 75 151 impbid ⊢ ( ( 𝐺 ∈ UMGraph ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) ) → ( ∃ 𝑓 ( 𝑓 ( Walks ‘ 𝐺 ) ⟨“ 𝐴 𝐵 𝐶 ”⟩ ∧ ( ♯ ‘ 𝑓 ) = 2 ) ↔ ( { 𝐴 , 𝐵 } ∈ 𝐸 ∧ { 𝐵 , 𝐶 } ∈ 𝐸 ) ) )
153 9 152 bitrd ⊢ ( ( 𝐺 ∈ UMGraph ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) ) → ( ⟨“ 𝐴 𝐵 𝐶 ”⟩ ∈ ( 𝐴 ( 2 WWalksNOn 𝐺 ) 𝐶 ) ↔ ( { 𝐴 , 𝐵 } ∈ 𝐸 ∧ { 𝐵 , 𝐶 } ∈ 𝐸 ) ) )