Step |
Hyp |
Ref |
Expression |
1 |
|
upgriswlk.v |
⊢ 𝑉 = ( Vtx ‘ 𝐺 ) |
2 |
|
upgriswlk.i |
⊢ 𝐼 = ( iEdg ‘ 𝐺 ) |
3 |
1 2
|
iswlkg |
⊢ ( 𝐺 ∈ UPGraph → ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ↔ ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) ) |
4 |
|
df-ifp |
⊢ ( if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) ↔ ( ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) ∧ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } ) ∨ ( ¬ ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) ∧ { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) |
5 |
|
dfsn2 |
⊢ { ( 𝑃 ‘ 𝑘 ) } = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ 𝑘 ) } |
6 |
|
preq2 |
⊢ ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) → { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ 𝑘 ) } = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ) |
7 |
5 6
|
syl5eq |
⊢ ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) → { ( 𝑃 ‘ 𝑘 ) } = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ) |
8 |
7
|
eqeq2d |
⊢ ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) → ( ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } ↔ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ) ) |
9 |
8
|
biimpa |
⊢ ( ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) ∧ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } ) → ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ) |
10 |
9
|
a1d |
⊢ ( ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) ∧ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } ) → ( ( ( 𝐺 ∈ UPGraph ∧ ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 ) ) ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ) ) |
11 |
|
eqid |
⊢ ( Edg ‘ 𝐺 ) = ( Edg ‘ 𝐺 ) |
12 |
2 11
|
upgredginwlk |
⊢ ( ( 𝐺 ∈ UPGraph ∧ 𝐹 ∈ Word dom 𝐼 ) → ( 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) → ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ( Edg ‘ 𝐺 ) ) ) |
13 |
12
|
adantrr |
⊢ ( ( 𝐺 ∈ UPGraph ∧ ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 ) ) → ( 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) → ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ( Edg ‘ 𝐺 ) ) ) |
14 |
13
|
imp |
⊢ ( ( ( 𝐺 ∈ UPGraph ∧ ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 ) ) ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ( Edg ‘ 𝐺 ) ) |
15 |
|
simp-4l |
⊢ ( ( ( ( ( 𝐺 ∈ UPGraph ∧ ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 ) ) ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ( Edg ‘ 𝐺 ) ) ∧ ( ¬ ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) ∧ { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) → 𝐺 ∈ UPGraph ) |
16 |
|
simpr |
⊢ ( ( ( ( 𝐺 ∈ UPGraph ∧ ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 ) ) ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ( Edg ‘ 𝐺 ) ) → ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ( Edg ‘ 𝐺 ) ) |
17 |
16
|
adantr |
⊢ ( ( ( ( ( 𝐺 ∈ UPGraph ∧ ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 ) ) ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ( Edg ‘ 𝐺 ) ) ∧ ( ¬ ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) ∧ { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) → ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ( Edg ‘ 𝐺 ) ) |
18 |
|
simpr |
⊢ ( ( ¬ ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) ∧ { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) → { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) |
19 |
18
|
adantl |
⊢ ( ( ( ( ( 𝐺 ∈ UPGraph ∧ ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 ) ) ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ( Edg ‘ 𝐺 ) ) ∧ ( ¬ ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) ∧ { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) → { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) |
20 |
|
fvexd |
⊢ ( ¬ ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) → ( 𝑃 ‘ 𝑘 ) ∈ V ) |
21 |
|
fvexd |
⊢ ( ¬ ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) → ( 𝑃 ‘ ( 𝑘 + 1 ) ) ∈ V ) |
22 |
|
neqne |
⊢ ( ¬ ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) → ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ ( 𝑘 + 1 ) ) ) |
23 |
20 21 22
|
3jca |
⊢ ( ¬ ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) → ( ( 𝑃 ‘ 𝑘 ) ∈ V ∧ ( 𝑃 ‘ ( 𝑘 + 1 ) ) ∈ V ∧ ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ ( 𝑘 + 1 ) ) ) ) |
24 |
23
|
adantr |
⊢ ( ( ¬ ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) ∧ { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) → ( ( 𝑃 ‘ 𝑘 ) ∈ V ∧ ( 𝑃 ‘ ( 𝑘 + 1 ) ) ∈ V ∧ ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ ( 𝑘 + 1 ) ) ) ) |
25 |
24
|
adantl |
⊢ ( ( ( ( ( 𝐺 ∈ UPGraph ∧ ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 ) ) ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ( Edg ‘ 𝐺 ) ) ∧ ( ¬ ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) ∧ { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) → ( ( 𝑃 ‘ 𝑘 ) ∈ V ∧ ( 𝑃 ‘ ( 𝑘 + 1 ) ) ∈ V ∧ ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ ( 𝑘 + 1 ) ) ) ) |
26 |
1 11
|
upgredgpr |
⊢ ( ( ( 𝐺 ∈ UPGraph ∧ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ( Edg ‘ 𝐺 ) ∧ { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) ∧ ( ( 𝑃 ‘ 𝑘 ) ∈ V ∧ ( 𝑃 ‘ ( 𝑘 + 1 ) ) ∈ V ∧ ( 𝑃 ‘ 𝑘 ) ≠ ( 𝑃 ‘ ( 𝑘 + 1 ) ) ) ) → { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } = ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) |
27 |
15 17 19 25 26
|
syl31anc |
⊢ ( ( ( ( ( 𝐺 ∈ UPGraph ∧ ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 ) ) ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ( Edg ‘ 𝐺 ) ) ∧ ( ¬ ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) ∧ { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) → { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } = ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) |
28 |
27
|
eqcomd |
⊢ ( ( ( ( ( 𝐺 ∈ UPGraph ∧ ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 ) ) ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ( Edg ‘ 𝐺 ) ) ∧ ( ¬ ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) ∧ { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) → ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ) |
29 |
28
|
exp31 |
⊢ ( ( ( 𝐺 ∈ UPGraph ∧ ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 ) ) ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ( Edg ‘ 𝐺 ) → ( ( ¬ ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) ∧ { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) → ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ) ) ) |
30 |
14 29
|
mpd |
⊢ ( ( ( 𝐺 ∈ UPGraph ∧ ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 ) ) ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( ¬ ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) ∧ { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) → ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ) ) |
31 |
30
|
com12 |
⊢ ( ( ¬ ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) ∧ { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) → ( ( ( 𝐺 ∈ UPGraph ∧ ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 ) ) ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ) ) |
32 |
10 31
|
jaoi |
⊢ ( ( ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) ∧ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } ) ∨ ( ¬ ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) ∧ { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) → ( ( ( 𝐺 ∈ UPGraph ∧ ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 ) ) ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ) ) |
33 |
32
|
com12 |
⊢ ( ( ( 𝐺 ∈ UPGraph ∧ ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 ) ) ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) ∧ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } ) ∨ ( ¬ ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) ∧ { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) → ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ) ) |
34 |
4 33
|
syl5bi |
⊢ ( ( ( 𝐺 ∈ UPGraph ∧ ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 ) ) ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) → ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ) ) |
35 |
|
ifpprsnss |
⊢ ( ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } → if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) |
36 |
34 35
|
impbid1 |
⊢ ( ( ( 𝐺 ∈ UPGraph ∧ ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 ) ) ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) ↔ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ) ) |
37 |
36
|
ralbidva |
⊢ ( ( 𝐺 ∈ UPGraph ∧ ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 ) ) → ( ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) ↔ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ) ) |
38 |
37
|
pm5.32da |
⊢ ( 𝐺 ∈ UPGraph → ( ( ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 ) ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ↔ ( ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 ) ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ) ) ) |
39 |
|
df-3an |
⊢ ( ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ↔ ( ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 ) ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) |
40 |
|
df-3an |
⊢ ( ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ) ↔ ( ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 ) ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ) ) |
41 |
38 39 40
|
3bitr4g |
⊢ ( 𝐺 ∈ UPGraph → ( ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ↔ ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ) ) ) |
42 |
3 41
|
bitrd |
⊢ ( 𝐺 ∈ UPGraph → ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ↔ ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ) ) ) |