Metamath Proof Explorer

Theorem wlkiswwlks1

Description: The sequence of vertices in a walk is a walk as word in a pseudograph. (Contributed by Alexander van der Vekens, 20-Jul-2018) (Revised by AV, 9-Apr-2021)

Ref Expression
Assertion wlkiswwlks1 ( 𝐺 ∈ UPGraph → ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → 𝑃 ∈ ( WWalks ‘ 𝐺 ) ) )

Proof

Step Hyp Ref Expression
1 wlkn0 ⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → 𝑃 ≠ ∅ )
2 eqid ⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 )
3 eqid ⊢ ( iEdg ‘ 𝐺 ) = ( iEdg ‘ 𝐺 )
4 2 3 upgriswlk ⊢ ( 𝐺 ∈ UPGraph → ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ↔ ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑖 ) ) = { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ) ) )
5 simpr ⊢ ( ( ( 𝐺 ∈ UPGraph ∧ ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑖 ) ) = { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ) ) ∧ 𝑃 ≠ ∅ ) → 𝑃 ≠ ∅ )
6 ffz0iswrd ⊢ ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) → 𝑃 ∈ Word ( Vtx ‘ 𝐺 ) )
7 6 3ad2ant2 ⊢ ( ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑖 ) ) = { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ) → 𝑃 ∈ Word ( Vtx ‘ 𝐺 ) )
8 7 ad2antlr ⊢ ( ( ( 𝐺 ∈ UPGraph ∧ ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑖 ) ) = { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ) ) ∧ 𝑃 ≠ ∅ ) → 𝑃 ∈ Word ( Vtx ‘ 𝐺 ) )
9 upgruhgr ⊢ ( 𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph )
10 3 uhgrfun ⊢ ( 𝐺 ∈ UHGraph → Fun ( iEdg ‘ 𝐺 ) )
11 funfn ⊢ ( Fun ( iEdg ‘ 𝐺 ) ↔ ( iEdg ‘ 𝐺 ) Fn dom ( iEdg ‘ 𝐺 ) )
12 11 biimpi ⊢ ( Fun ( iEdg ‘ 𝐺 ) → ( iEdg ‘ 𝐺 ) Fn dom ( iEdg ‘ 𝐺 ) )
13 9 10 12 3syl ⊢ ( 𝐺 ∈ UPGraph → ( iEdg ‘ 𝐺 ) Fn dom ( iEdg ‘ 𝐺 ) )
14 13 ad2antlr ⊢ ( ( ( ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ∧ 𝐺 ∈ UPGraph ) ∧ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( iEdg ‘ 𝐺 ) Fn dom ( iEdg ‘ 𝐺 ) )
15 wrdsymbcl ⊢ ( ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( 𝐹 ‘ 𝑖 ) ∈ dom ( iEdg ‘ 𝐺 ) )
16 15 ad4ant14 ⊢ ( ( ( ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ∧ 𝐺 ∈ UPGraph ) ∧ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( 𝐹 ‘ 𝑖 ) ∈ dom ( iEdg ‘ 𝐺 ) )
17 fnfvelrn ⊢ ( ( ( iEdg ‘ 𝐺 ) Fn dom ( iEdg ‘ 𝐺 ) ∧ ( 𝐹 ‘ 𝑖 ) ∈ dom ( iEdg ‘ 𝐺 ) ) → ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑖 ) ) ∈ ran ( iEdg ‘ 𝐺 ) )
18 14 16 17 syl2anc ⊢ ( ( ( ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ∧ 𝐺 ∈ UPGraph ) ∧ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑖 ) ) ∈ ran ( iEdg ‘ 𝐺 ) )
19 edgval ⊢ ( Edg ‘ 𝐺 ) = ran ( iEdg ‘ 𝐺 )
20 18 19 eleqtrrdi ⊢ ( ( ( ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ∧ 𝐺 ∈ UPGraph ) ∧ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑖 ) ) ∈ ( Edg ‘ 𝐺 ) )
21 eleq1 ⊢ ( { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } = ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑖 ) ) → ( { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ↔ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑖 ) ) ∈ ( Edg ‘ 𝐺 ) ) )
22 21 eqcoms ⊢ ( ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑖 ) ) = { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } → ( { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ↔ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑖 ) ) ∈ ( Edg ‘ 𝐺 ) ) )
23 20 22 syl5ibrcom ⊢ ( ( ( ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ∧ 𝐺 ∈ UPGraph ) ∧ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑖 ) ) = { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } → { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ) )
24 23 ralimdva ⊢ ( ( ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ∧ 𝐺 ∈ UPGraph ) → ( ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑖 ) ) = { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } → ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ) )
25 24 ex ⊢ ( ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) → ( 𝐺 ∈ UPGraph → ( ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑖 ) ) = { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } → ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ) ) )
26 25 com23 ⊢ ( ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) → ( ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑖 ) ) = { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } → ( 𝐺 ∈ UPGraph → ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ) ) )
27 26 3impia ⊢ ( ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑖 ) ) = { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ) → ( 𝐺 ∈ UPGraph → ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ) )
28 27 impcom ⊢ ( ( 𝐺 ∈ UPGraph ∧ ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑖 ) ) = { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ) ) → ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) )
29 lencl ⊢ ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) → ( ♯ ‘ 𝐹 ) ∈ ℕ0 )
30 ffz0hash ⊢ ( ( ( ♯ ‘ 𝐹 ) ∈ ℕ0 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) → ( ♯ ‘ 𝑃 ) = ( ( ♯ ‘ 𝐹 ) + 1 ) )
31 30 ex ⊢ ( ( ♯ ‘ 𝐹 ) ∈ ℕ0 → ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) → ( ♯ ‘ 𝑃 ) = ( ( ♯ ‘ 𝐹 ) + 1 ) ) )
32 oveq1 ⊢ ( ( ♯ ‘ 𝑃 ) = ( ( ♯ ‘ 𝐹 ) + 1 ) → ( ( ♯ ‘ 𝑃 ) − 1 ) = ( ( ( ♯ ‘ 𝐹 ) + 1 ) − 1 ) )
33 nn0cn ⊢ ( ( ♯ ‘ 𝐹 ) ∈ ℕ0 → ( ♯ ‘ 𝐹 ) ∈ ℂ )
34 pncan1 ⊢ ( ( ♯ ‘ 𝐹 ) ∈ ℂ → ( ( ( ♯ ‘ 𝐹 ) + 1 ) − 1 ) = ( ♯ ‘ 𝐹 ) )
35 33 34 syl ⊢ ( ( ♯ ‘ 𝐹 ) ∈ ℕ0 → ( ( ( ♯ ‘ 𝐹 ) + 1 ) − 1 ) = ( ♯ ‘ 𝐹 ) )
36 32 35 sylan9eqr ⊢ ( ( ( ♯ ‘ 𝐹 ) ∈ ℕ0 ∧ ( ♯ ‘ 𝑃 ) = ( ( ♯ ‘ 𝐹 ) + 1 ) ) → ( ( ♯ ‘ 𝑃 ) − 1 ) = ( ♯ ‘ 𝐹 ) )
37 36 ex ⊢ ( ( ♯ ‘ 𝐹 ) ∈ ℕ0 → ( ( ♯ ‘ 𝑃 ) = ( ( ♯ ‘ 𝐹 ) + 1 ) → ( ( ♯ ‘ 𝑃 ) − 1 ) = ( ♯ ‘ 𝐹 ) ) )
38 31 37 syld ⊢ ( ( ♯ ‘ 𝐹 ) ∈ ℕ0 → ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) → ( ( ♯ ‘ 𝑃 ) − 1 ) = ( ♯ ‘ 𝐹 ) ) )
39 29 38 syl ⊢ ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) → ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) → ( ( ♯ ‘ 𝑃 ) − 1 ) = ( ♯ ‘ 𝐹 ) ) )
40 39 imp ⊢ ( ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) → ( ( ♯ ‘ 𝑃 ) − 1 ) = ( ♯ ‘ 𝐹 ) )
41 40 oveq2d ⊢ ( ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) → ( 0 ..^ ( ( ♯ ‘ 𝑃 ) − 1 ) ) = ( 0 ..^ ( ♯ ‘ 𝐹 ) ) )
42 41 raleqdv ⊢ ( ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) → ( ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑃 ) − 1 ) ) { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ↔ ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ) )
43 42 3adant3 ⊢ ( ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑖 ) ) = { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ) → ( ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑃 ) − 1 ) ) { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ↔ ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ) )
44 43 adantl ⊢ ( ( 𝐺 ∈ UPGraph ∧ ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑖 ) ) = { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ) ) → ( ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑃 ) − 1 ) ) { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ↔ ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ) )
45 28 44 mpbird ⊢ ( ( 𝐺 ∈ UPGraph ∧ ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑖 ) ) = { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ) ) → ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑃 ) − 1 ) ) { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) )
46 45 adantr ⊢ ( ( ( 𝐺 ∈ UPGraph ∧ ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑖 ) ) = { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ) ) ∧ 𝑃 ≠ ∅ ) → ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑃 ) − 1 ) ) { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) )
47 eqid ⊢ ( Edg ‘ 𝐺 ) = ( Edg ‘ 𝐺 )
48 2 47 iswwlks ⊢ ( 𝑃 ∈ ( WWalks ‘ 𝐺 ) ↔ ( 𝑃 ≠ ∅ ∧ 𝑃 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑃 ) − 1 ) ) { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ) )
49 5 8 46 48 syl3anbrc ⊢ ( ( ( 𝐺 ∈ UPGraph ∧ ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑖 ) ) = { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ) ) ∧ 𝑃 ≠ ∅ ) → 𝑃 ∈ ( WWalks ‘ 𝐺 ) )
50 49 ex ⊢ ( ( 𝐺 ∈ UPGraph ∧ ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑖 ) ) = { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ) ) → ( 𝑃 ≠ ∅ → 𝑃 ∈ ( WWalks ‘ 𝐺 ) ) )
51 50 ex ⊢ ( 𝐺 ∈ UPGraph → ( ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑖 ) ) = { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ) → ( 𝑃 ≠ ∅ → 𝑃 ∈ ( WWalks ‘ 𝐺 ) ) ) )
52 4 51 sylbid ⊢ ( 𝐺 ∈ UPGraph → ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( 𝑃 ≠ ∅ → 𝑃 ∈ ( WWalks ‘ 𝐺 ) ) ) )
53 1 52 mpdi ⊢ ( 𝐺 ∈ UPGraph → ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → 𝑃 ∈ ( WWalks ‘ 𝐺 ) ) )