Step |
Hyp |
Ref |
Expression |
1 |
|
wlkv |
⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( 𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V ) ) |
2 |
|
simp3l |
⊢ ( ( 𝐺 ∈ V ∧ 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( Fun ( iEdg ‘ 𝐺 ) ∧ ( ( ♯ ‘ 𝐹 ) = 1 ∧ ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ 1 ) ) ) ) → Fun ( iEdg ‘ 𝐺 ) ) |
3 |
|
simp2 |
⊢ ( ( 𝐺 ∈ V ∧ 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( Fun ( iEdg ‘ 𝐺 ) ∧ ( ( ♯ ‘ 𝐹 ) = 1 ∧ ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ 1 ) ) ) ) → 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ) |
4 |
|
c0ex |
⊢ 0 ∈ V |
5 |
4
|
snid |
⊢ 0 ∈ { 0 } |
6 |
|
oveq2 |
⊢ ( ( ♯ ‘ 𝐹 ) = 1 → ( 0 ..^ ( ♯ ‘ 𝐹 ) ) = ( 0 ..^ 1 ) ) |
7 |
|
fzo01 |
⊢ ( 0 ..^ 1 ) = { 0 } |
8 |
6 7
|
eqtrdi |
⊢ ( ( ♯ ‘ 𝐹 ) = 1 → ( 0 ..^ ( ♯ ‘ 𝐹 ) ) = { 0 } ) |
9 |
5 8
|
eleqtrrid |
⊢ ( ( ♯ ‘ 𝐹 ) = 1 → 0 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) |
10 |
9
|
ad2antrl |
⊢ ( ( Fun ( iEdg ‘ 𝐺 ) ∧ ( ( ♯ ‘ 𝐹 ) = 1 ∧ ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ 1 ) ) ) → 0 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) |
11 |
10
|
3ad2ant3 |
⊢ ( ( 𝐺 ∈ V ∧ 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( Fun ( iEdg ‘ 𝐺 ) ∧ ( ( ♯ ‘ 𝐹 ) = 1 ∧ ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ 1 ) ) ) ) → 0 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) |
12 |
|
eqid |
⊢ ( iEdg ‘ 𝐺 ) = ( iEdg ‘ 𝐺 ) |
13 |
12
|
iedginwlk |
⊢ ( ( Fun ( iEdg ‘ 𝐺 ) ∧ 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ 0 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 0 ) ) ∈ ran ( iEdg ‘ 𝐺 ) ) |
14 |
2 3 11 13
|
syl3anc |
⊢ ( ( 𝐺 ∈ V ∧ 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( Fun ( iEdg ‘ 𝐺 ) ∧ ( ( ♯ ‘ 𝐹 ) = 1 ∧ ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ 1 ) ) ) ) → ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 0 ) ) ∈ ran ( iEdg ‘ 𝐺 ) ) |
15 |
|
eqid |
⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 ) |
16 |
15 12
|
iswlkg |
⊢ ( 𝐺 ∈ V → ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ↔ ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) ) |
17 |
8
|
raleqdv |
⊢ ( ( ♯ ‘ 𝐹 ) = 1 → ( ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) ↔ ∀ 𝑘 ∈ { 0 } if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) |
18 |
|
oveq1 |
⊢ ( 𝑘 = 0 → ( 𝑘 + 1 ) = ( 0 + 1 ) ) |
19 |
|
0p1e1 |
⊢ ( 0 + 1 ) = 1 |
20 |
18 19
|
eqtrdi |
⊢ ( 𝑘 = 0 → ( 𝑘 + 1 ) = 1 ) |
21 |
|
wkslem2 |
⊢ ( ( 𝑘 = 0 ∧ ( 𝑘 + 1 ) = 1 ) → ( if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) ↔ if- ( ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ 1 ) , ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 0 ) ) = { ( 𝑃 ‘ 0 ) } , { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 0 ) ) ) ) ) |
22 |
20 21
|
mpdan |
⊢ ( 𝑘 = 0 → ( if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) ↔ if- ( ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ 1 ) , ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 0 ) ) = { ( 𝑃 ‘ 0 ) } , { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 0 ) ) ) ) ) |
23 |
4 22
|
ralsn |
⊢ ( ∀ 𝑘 ∈ { 0 } if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) ↔ if- ( ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ 1 ) , ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 0 ) ) = { ( 𝑃 ‘ 0 ) } , { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 0 ) ) ) ) |
24 |
17 23
|
bitrdi |
⊢ ( ( ♯ ‘ 𝐹 ) = 1 → ( ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) ↔ if- ( ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ 1 ) , ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 0 ) ) = { ( 𝑃 ‘ 0 ) } , { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 0 ) ) ) ) ) |
25 |
24
|
ad2antrl |
⊢ ( ( Fun ( iEdg ‘ 𝐺 ) ∧ ( ( ♯ ‘ 𝐹 ) = 1 ∧ ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ 1 ) ) ) → ( ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) ↔ if- ( ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ 1 ) , ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 0 ) ) = { ( 𝑃 ‘ 0 ) } , { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 0 ) ) ) ) ) |
26 |
|
ifptru |
⊢ ( ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ 1 ) → ( if- ( ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ 1 ) , ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 0 ) ) = { ( 𝑃 ‘ 0 ) } , { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 0 ) ) ) ↔ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 0 ) ) = { ( 𝑃 ‘ 0 ) } ) ) |
27 |
26
|
biimpa |
⊢ ( ( ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ 1 ) ∧ if- ( ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ 1 ) , ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 0 ) ) = { ( 𝑃 ‘ 0 ) } , { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 0 ) ) ) ) → ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 0 ) ) = { ( 𝑃 ‘ 0 ) } ) |
28 |
27
|
eqcomd |
⊢ ( ( ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ 1 ) ∧ if- ( ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ 1 ) , ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 0 ) ) = { ( 𝑃 ‘ 0 ) } , { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 0 ) ) ) ) → { ( 𝑃 ‘ 0 ) } = ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 0 ) ) ) |
29 |
28
|
ex |
⊢ ( ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ 1 ) → ( if- ( ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ 1 ) , ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 0 ) ) = { ( 𝑃 ‘ 0 ) } , { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 0 ) ) ) → { ( 𝑃 ‘ 0 ) } = ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 0 ) ) ) ) |
30 |
29
|
ad2antll |
⊢ ( ( Fun ( iEdg ‘ 𝐺 ) ∧ ( ( ♯ ‘ 𝐹 ) = 1 ∧ ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ 1 ) ) ) → ( if- ( ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ 1 ) , ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 0 ) ) = { ( 𝑃 ‘ 0 ) } , { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 0 ) ) ) → { ( 𝑃 ‘ 0 ) } = ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 0 ) ) ) ) |
31 |
25 30
|
sylbid |
⊢ ( ( Fun ( iEdg ‘ 𝐺 ) ∧ ( ( ♯ ‘ 𝐹 ) = 1 ∧ ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ 1 ) ) ) → ( ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) → { ( 𝑃 ‘ 0 ) } = ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 0 ) ) ) ) |
32 |
31
|
com12 |
⊢ ( ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) → ( ( Fun ( iEdg ‘ 𝐺 ) ∧ ( ( ♯ ‘ 𝐹 ) = 1 ∧ ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ 1 ) ) ) → { ( 𝑃 ‘ 0 ) } = ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 0 ) ) ) ) |
33 |
32
|
3ad2ant3 |
⊢ ( ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) → ( ( Fun ( iEdg ‘ 𝐺 ) ∧ ( ( ♯ ‘ 𝐹 ) = 1 ∧ ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ 1 ) ) ) → { ( 𝑃 ‘ 0 ) } = ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 0 ) ) ) ) |
34 |
16 33
|
syl6bi |
⊢ ( 𝐺 ∈ V → ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( ( Fun ( iEdg ‘ 𝐺 ) ∧ ( ( ♯ ‘ 𝐹 ) = 1 ∧ ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ 1 ) ) ) → { ( 𝑃 ‘ 0 ) } = ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 0 ) ) ) ) ) |
35 |
34
|
3imp |
⊢ ( ( 𝐺 ∈ V ∧ 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( Fun ( iEdg ‘ 𝐺 ) ∧ ( ( ♯ ‘ 𝐹 ) = 1 ∧ ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ 1 ) ) ) ) → { ( 𝑃 ‘ 0 ) } = ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 0 ) ) ) |
36 |
|
edgval |
⊢ ( Edg ‘ 𝐺 ) = ran ( iEdg ‘ 𝐺 ) |
37 |
36
|
a1i |
⊢ ( ( 𝐺 ∈ V ∧ 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( Fun ( iEdg ‘ 𝐺 ) ∧ ( ( ♯ ‘ 𝐹 ) = 1 ∧ ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ 1 ) ) ) ) → ( Edg ‘ 𝐺 ) = ran ( iEdg ‘ 𝐺 ) ) |
38 |
14 35 37
|
3eltr4d |
⊢ ( ( 𝐺 ∈ V ∧ 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ ( Fun ( iEdg ‘ 𝐺 ) ∧ ( ( ♯ ‘ 𝐹 ) = 1 ∧ ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ 1 ) ) ) ) → { ( 𝑃 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) ) |
39 |
38
|
3exp |
⊢ ( 𝐺 ∈ V → ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( ( Fun ( iEdg ‘ 𝐺 ) ∧ ( ( ♯ ‘ 𝐹 ) = 1 ∧ ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ 1 ) ) ) → { ( 𝑃 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) ) ) ) |
40 |
39
|
3ad2ant1 |
⊢ ( ( 𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V ) → ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( ( Fun ( iEdg ‘ 𝐺 ) ∧ ( ( ♯ ‘ 𝐹 ) = 1 ∧ ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ 1 ) ) ) → { ( 𝑃 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) ) ) ) |
41 |
1 40
|
mpcom |
⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( ( Fun ( iEdg ‘ 𝐺 ) ∧ ( ( ♯ ‘ 𝐹 ) = 1 ∧ ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ 1 ) ) ) → { ( 𝑃 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) ) ) |
42 |
41
|
expd |
⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( Fun ( iEdg ‘ 𝐺 ) → ( ( ( ♯ ‘ 𝐹 ) = 1 ∧ ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ 1 ) ) → { ( 𝑃 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) ) ) ) |
43 |
42
|
impcom |
⊢ ( ( Fun ( iEdg ‘ 𝐺 ) ∧ 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ) → ( ( ( ♯ ‘ 𝐹 ) = 1 ∧ ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ 1 ) ) → { ( 𝑃 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) ) ) |
44 |
43
|
imp |
⊢ ( ( ( Fun ( iEdg ‘ 𝐺 ) ∧ 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ) ∧ ( ( ♯ ‘ 𝐹 ) = 1 ∧ ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ 1 ) ) ) → { ( 𝑃 ‘ 0 ) } ∈ ( Edg ‘ 𝐺 ) ) |