| Step |
Hyp |
Ref |
Expression |
| 1 |
|
subgrv |
⊢ ( 𝑆 SubGraph 𝐺 → ( 𝑆 ∈ V ∧ 𝐺 ∈ V ) ) |
| 2 |
1
|
simpld |
⊢ ( 𝑆 SubGraph 𝐺 → 𝑆 ∈ V ) |
| 3 |
|
eqid |
⊢ ( Vtx ‘ 𝑆 ) = ( Vtx ‘ 𝑆 ) |
| 4 |
|
eqid |
⊢ ( iEdg ‘ 𝑆 ) = ( iEdg ‘ 𝑆 ) |
| 5 |
3 4
|
iswlkg |
⊢ ( 𝑆 ∈ V → ( 𝐹 ( Walks ‘ 𝑆 ) 𝑃 ↔ ( 𝐹 ∈ Word dom ( iEdg ‘ 𝑆 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝑆 ) ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( ( iEdg ‘ 𝑆 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝑆 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) ) |
| 6 |
2 5
|
syl |
⊢ ( 𝑆 SubGraph 𝐺 → ( 𝐹 ( Walks ‘ 𝑆 ) 𝑃 ↔ ( 𝐹 ∈ Word dom ( iEdg ‘ 𝑆 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝑆 ) ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( ( iEdg ‘ 𝑆 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝑆 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) ) |
| 7 |
|
3simpa |
⊢ ( ( 𝐹 ∈ Word dom ( iEdg ‘ 𝑆 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝑆 ) ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( ( iEdg ‘ 𝑆 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝑆 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) → ( 𝐹 ∈ Word dom ( iEdg ‘ 𝑆 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝑆 ) ) ) |
| 8 |
|
eqid |
⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 ) |
| 9 |
|
eqid |
⊢ ( iEdg ‘ 𝐺 ) = ( iEdg ‘ 𝐺 ) |
| 10 |
|
eqid |
⊢ ( Edg ‘ 𝑆 ) = ( Edg ‘ 𝑆 ) |
| 11 |
3 8 4 9 10
|
subgrprop2 |
⊢ ( 𝑆 SubGraph 𝐺 → ( ( Vtx ‘ 𝑆 ) ⊆ ( Vtx ‘ 𝐺 ) ∧ ( iEdg ‘ 𝑆 ) ⊆ ( iEdg ‘ 𝐺 ) ∧ ( Edg ‘ 𝑆 ) ⊆ 𝒫 ( Vtx ‘ 𝑆 ) ) ) |
| 12 |
11
|
simp2d |
⊢ ( 𝑆 SubGraph 𝐺 → ( iEdg ‘ 𝑆 ) ⊆ ( iEdg ‘ 𝐺 ) ) |
| 13 |
|
dmss |
⊢ ( ( iEdg ‘ 𝑆 ) ⊆ ( iEdg ‘ 𝐺 ) → dom ( iEdg ‘ 𝑆 ) ⊆ dom ( iEdg ‘ 𝐺 ) ) |
| 14 |
|
sswrd |
⊢ ( dom ( iEdg ‘ 𝑆 ) ⊆ dom ( iEdg ‘ 𝐺 ) → Word dom ( iEdg ‘ 𝑆 ) ⊆ Word dom ( iEdg ‘ 𝐺 ) ) |
| 15 |
12 13 14
|
3syl |
⊢ ( 𝑆 SubGraph 𝐺 → Word dom ( iEdg ‘ 𝑆 ) ⊆ Word dom ( iEdg ‘ 𝐺 ) ) |
| 16 |
15
|
sseld |
⊢ ( 𝑆 SubGraph 𝐺 → ( 𝐹 ∈ Word dom ( iEdg ‘ 𝑆 ) → 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ) ) |
| 17 |
11
|
simp1d |
⊢ ( 𝑆 SubGraph 𝐺 → ( Vtx ‘ 𝑆 ) ⊆ ( Vtx ‘ 𝐺 ) ) |
| 18 |
|
fss |
⊢ ( ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝑆 ) ∧ ( Vtx ‘ 𝑆 ) ⊆ ( Vtx ‘ 𝐺 ) ) → 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) |
| 19 |
18
|
expcom |
⊢ ( ( Vtx ‘ 𝑆 ) ⊆ ( Vtx ‘ 𝐺 ) → ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝑆 ) → 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ) |
| 20 |
17 19
|
syl |
⊢ ( 𝑆 SubGraph 𝐺 → ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝑆 ) → 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ) |
| 21 |
16 20
|
anim12d |
⊢ ( 𝑆 SubGraph 𝐺 → ( ( 𝐹 ∈ Word dom ( iEdg ‘ 𝑆 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝑆 ) ) → ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ) ) |
| 22 |
7 21
|
syl5 |
⊢ ( 𝑆 SubGraph 𝐺 → ( ( 𝐹 ∈ Word dom ( iEdg ‘ 𝑆 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝑆 ) ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( ( iEdg ‘ 𝑆 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝑆 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) → ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ) ) |
| 23 |
|
3simpb |
⊢ ( ( 𝐹 ∈ Word dom ( iEdg ‘ 𝑆 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝑆 ) ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( ( iEdg ‘ 𝑆 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝑆 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) → ( 𝐹 ∈ Word dom ( iEdg ‘ 𝑆 ) ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( ( iEdg ‘ 𝑆 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝑆 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) |
| 24 |
3 8 4 9 10
|
subgrprop |
⊢ ( 𝑆 SubGraph 𝐺 → ( ( Vtx ‘ 𝑆 ) ⊆ ( Vtx ‘ 𝐺 ) ∧ ( iEdg ‘ 𝑆 ) = ( ( iEdg ‘ 𝐺 ) ↾ dom ( iEdg ‘ 𝑆 ) ) ∧ ( Edg ‘ 𝑆 ) ⊆ 𝒫 ( Vtx ‘ 𝑆 ) ) ) |
| 25 |
24
|
simp2d |
⊢ ( 𝑆 SubGraph 𝐺 → ( iEdg ‘ 𝑆 ) = ( ( iEdg ‘ 𝐺 ) ↾ dom ( iEdg ‘ 𝑆 ) ) ) |
| 26 |
25
|
fveq1d |
⊢ ( 𝑆 SubGraph 𝐺 → ( ( iEdg ‘ 𝑆 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = ( ( ( iEdg ‘ 𝐺 ) ↾ dom ( iEdg ‘ 𝑆 ) ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) |
| 27 |
26
|
3ad2ant1 |
⊢ ( ( 𝑆 SubGraph 𝐺 ∧ 𝐹 ∈ Word dom ( iEdg ‘ 𝑆 ) ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( iEdg ‘ 𝑆 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = ( ( ( iEdg ‘ 𝐺 ) ↾ dom ( iEdg ‘ 𝑆 ) ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) |
| 28 |
|
wrdsymbcl |
⊢ ( ( 𝐹 ∈ Word dom ( iEdg ‘ 𝑆 ) ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( 𝐹 ‘ 𝑘 ) ∈ dom ( iEdg ‘ 𝑆 ) ) |
| 29 |
28
|
fvresd |
⊢ ( ( 𝐹 ∈ Word dom ( iEdg ‘ 𝑆 ) ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( ( iEdg ‘ 𝐺 ) ↾ dom ( iEdg ‘ 𝑆 ) ) ‘ ( 𝐹 ‘ 𝑘 ) ) = ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) |
| 30 |
29
|
3adant1 |
⊢ ( ( 𝑆 SubGraph 𝐺 ∧ 𝐹 ∈ Word dom ( iEdg ‘ 𝑆 ) ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( ( iEdg ‘ 𝐺 ) ↾ dom ( iEdg ‘ 𝑆 ) ) ‘ ( 𝐹 ‘ 𝑘 ) ) = ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) |
| 31 |
27 30
|
eqtrd |
⊢ ( ( 𝑆 SubGraph 𝐺 ∧ 𝐹 ∈ Word dom ( iEdg ‘ 𝑆 ) ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( iEdg ‘ 𝑆 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) |
| 32 |
31
|
eqeq1d |
⊢ ( ( 𝑆 SubGraph 𝐺 ∧ 𝐹 ∈ Word dom ( iEdg ‘ 𝑆 ) ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( ( iEdg ‘ 𝑆 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } ↔ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } ) ) |
| 33 |
31
|
sseq2d |
⊢ ( ( 𝑆 SubGraph 𝐺 ∧ 𝐹 ∈ Word dom ( iEdg ‘ 𝑆 ) ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝑆 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ↔ { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) |
| 34 |
32 33
|
ifpbi23d |
⊢ ( ( 𝑆 SubGraph 𝐺 ∧ 𝐹 ∈ Word dom ( iEdg ‘ 𝑆 ) ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( ( iEdg ‘ 𝑆 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝑆 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) ↔ if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) |
| 35 |
34
|
biimpd |
⊢ ( ( 𝑆 SubGraph 𝐺 ∧ 𝐹 ∈ Word dom ( iEdg ‘ 𝑆 ) ∧ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( ( iEdg ‘ 𝑆 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝑆 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) → if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) |
| 36 |
35
|
3expia |
⊢ ( ( 𝑆 SubGraph 𝐺 ∧ 𝐹 ∈ Word dom ( iEdg ‘ 𝑆 ) ) → ( 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) → ( if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( ( iEdg ‘ 𝑆 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝑆 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) → if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) ) |
| 37 |
36
|
ralrimiv |
⊢ ( ( 𝑆 SubGraph 𝐺 ∧ 𝐹 ∈ Word dom ( iEdg ‘ 𝑆 ) ) → ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( ( iEdg ‘ 𝑆 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝑆 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) → if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) |
| 38 |
|
ralim |
⊢ ( ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( ( iEdg ‘ 𝑆 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝑆 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) → if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) → ( ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( ( iEdg ‘ 𝑆 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝑆 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) → ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) |
| 39 |
37 38
|
syl |
⊢ ( ( 𝑆 SubGraph 𝐺 ∧ 𝐹 ∈ Word dom ( iEdg ‘ 𝑆 ) ) → ( ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( ( iEdg ‘ 𝑆 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝑆 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) → ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) |
| 40 |
39
|
expimpd |
⊢ ( 𝑆 SubGraph 𝐺 → ( ( 𝐹 ∈ Word dom ( iEdg ‘ 𝑆 ) ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( ( iEdg ‘ 𝑆 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝑆 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) → ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) |
| 41 |
23 40
|
syl5 |
⊢ ( 𝑆 SubGraph 𝐺 → ( ( 𝐹 ∈ Word dom ( iEdg ‘ 𝑆 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝑆 ) ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( ( iEdg ‘ 𝑆 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝑆 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) → ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) |
| 42 |
22 41
|
jcad |
⊢ ( 𝑆 SubGraph 𝐺 → ( ( 𝐹 ∈ Word dom ( iEdg ‘ 𝑆 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝑆 ) ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑘 ) = ( 𝑃 ‘ ( 𝑘 + 1 ) ) , ( ( iEdg ‘ 𝑆 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) } , { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝑆 ) ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) → ( ( 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 ) ) ) ) ) ) |
| 43 |
6 42
|
sylbid |
⊢ ( 𝑆 SubGraph 𝐺 → ( 𝐹 ( Walks ‘ 𝑆 ) 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 ) ) ) ) ) ) |
| 44 |
|
df-3an |
⊢ ( ( 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 ) ) ) ) ↔ ( ( 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 ) ) ) ) ) |
| 45 |
43 44
|
imbitrrdi |
⊢ ( 𝑆 SubGraph 𝐺 → ( 𝐹 ( Walks ‘ 𝑆 ) 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 ) ) ) ) ) ) |
| 46 |
8 9
|
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 ) ) ) ) ) ) |
| 47 |
1 46
|
simpl2im |
⊢ ( 𝑆 SubGraph 𝐺 → ( 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 ) ) ) ) ) ) |
| 48 |
45 47
|
sylibrd |
⊢ ( 𝑆 SubGraph 𝐺 → ( 𝐹 ( Walks ‘ 𝑆 ) 𝑃 → 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ) ) |