Step |
Hyp |
Ref |
Expression |
1 |
|
wksfval.v |
⊢ 𝑉 = ( Vtx ‘ 𝐺 ) |
2 |
|
wksfval.i |
⊢ 𝐼 = ( iEdg ‘ 𝐺 ) |
3 |
|
df-wlks |
⊢ Walks = ( 𝑔 ∈ V ↦ { 〈 𝑓 , 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 ) ) ) ) } ) |
4 |
|
fveq2 |
⊢ ( 𝑔 = 𝐺 → ( iEdg ‘ 𝑔 ) = ( iEdg ‘ 𝐺 ) ) |
5 |
4 2
|
eqtr4di |
⊢ ( 𝑔 = 𝐺 → ( iEdg ‘ 𝑔 ) = 𝐼 ) |
6 |
5
|
dmeqd |
⊢ ( 𝑔 = 𝐺 → dom ( iEdg ‘ 𝑔 ) = dom 𝐼 ) |
7 |
|
wrdeq |
⊢ ( dom ( iEdg ‘ 𝑔 ) = dom 𝐼 → Word dom ( iEdg ‘ 𝑔 ) = Word dom 𝐼 ) |
8 |
6 7
|
syl |
⊢ ( 𝑔 = 𝐺 → Word dom ( iEdg ‘ 𝑔 ) = Word dom 𝐼 ) |
9 |
8
|
eleq2d |
⊢ ( 𝑔 = 𝐺 → ( 𝑓 ∈ Word dom ( iEdg ‘ 𝑔 ) ↔ 𝑓 ∈ Word dom 𝐼 ) ) |
10 |
|
fveq2 |
⊢ ( 𝑔 = 𝐺 → ( Vtx ‘ 𝑔 ) = ( Vtx ‘ 𝐺 ) ) |
11 |
10 1
|
eqtr4di |
⊢ ( 𝑔 = 𝐺 → ( Vtx ‘ 𝑔 ) = 𝑉 ) |
12 |
11
|
feq3d |
⊢ ( 𝑔 = 𝐺 → ( 𝑝 : ( 0 ... ( ♯ ‘ 𝑓 ) ) ⟶ ( Vtx ‘ 𝑔 ) ↔ 𝑝 : ( 0 ... ( ♯ ‘ 𝑓 ) ) ⟶ 𝑉 ) ) |
13 |
5
|
fveq1d |
⊢ ( 𝑔 = 𝐺 → ( ( iEdg ‘ 𝑔 ) ‘ ( 𝑓 ‘ 𝑘 ) ) = ( 𝐼 ‘ ( 𝑓 ‘ 𝑘 ) ) ) |
14 |
13
|
eqeq1d |
⊢ ( 𝑔 = 𝐺 → ( ( ( iEdg ‘ 𝑔 ) ‘ ( 𝑓 ‘ 𝑘 ) ) = { ( 𝑝 ‘ 𝑘 ) } ↔ ( 𝐼 ‘ ( 𝑓 ‘ 𝑘 ) ) = { ( 𝑝 ‘ 𝑘 ) } ) ) |
15 |
13
|
sseq2d |
⊢ ( 𝑔 = 𝐺 → ( { ( 𝑝 ‘ 𝑘 ) , ( 𝑝 ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝑔 ) ‘ ( 𝑓 ‘ 𝑘 ) ) ↔ { ( 𝑝 ‘ 𝑘 ) , ( 𝑝 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝑓 ‘ 𝑘 ) ) ) ) |
16 |
14 15
|
ifpbi23d |
⊢ ( 𝑔 = 𝐺 → ( if- ( ( 𝑝 ‘ 𝑘 ) = ( 𝑝 ‘ ( 𝑘 + 1 ) ) , ( ( iEdg ‘ 𝑔 ) ‘ ( 𝑓 ‘ 𝑘 ) ) = { ( 𝑝 ‘ 𝑘 ) } , { ( 𝑝 ‘ 𝑘 ) , ( 𝑝 ‘ ( 𝑘 + 1 ) ) } ⊆ ( ( iEdg ‘ 𝑔 ) ‘ ( 𝑓 ‘ 𝑘 ) ) ) ↔ if- ( ( 𝑝 ‘ 𝑘 ) = ( 𝑝 ‘ ( 𝑘 + 1 ) ) , ( 𝐼 ‘ ( 𝑓 ‘ 𝑘 ) ) = { ( 𝑝 ‘ 𝑘 ) } , { ( 𝑝 ‘ 𝑘 ) , ( 𝑝 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝑓 ‘ 𝑘 ) ) ) ) ) |
17 |
16
|
ralbidv |
⊢ ( 𝑔 = 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 ) ) ) ↔ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝑓 ) ) if- ( ( 𝑝 ‘ 𝑘 ) = ( 𝑝 ‘ ( 𝑘 + 1 ) ) , ( 𝐼 ‘ ( 𝑓 ‘ 𝑘 ) ) = { ( 𝑝 ‘ 𝑘 ) } , { ( 𝑝 ‘ 𝑘 ) , ( 𝑝 ‘ ( 𝑘 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝑓 ‘ 𝑘 ) ) ) ) ) |
18 |
9 12 17
|
3anbi123d |
⊢ ( 𝑔 = 𝐺 → ( ( 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 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 ) ) ) ) ) ) |
19 |
18
|
opabbidv |
⊢ ( 𝑔 = 𝐺 → { 〈 𝑓 , 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 ) ) ) ) } = { 〈 𝑓 , 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 ) ) ) ) } ) |
20 |
|
elex |
⊢ ( 𝐺 ∈ 𝑊 → 𝐺 ∈ V ) |
21 |
|
3anass |
⊢ ( ( 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 ) ) ) ) ) ) |
22 |
21
|
opabbii |
⊢ { 〈 𝑓 , 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 ) ) ) ) } = { 〈 𝑓 , 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 ) ) ) ) ) } |
23 |
2
|
fvexi |
⊢ 𝐼 ∈ V |
24 |
23
|
dmex |
⊢ dom 𝐼 ∈ V |
25 |
|
wrdexg |
⊢ ( dom 𝐼 ∈ V → Word dom 𝐼 ∈ V ) |
26 |
24 25
|
mp1i |
⊢ ( 𝐺 ∈ 𝑊 → Word dom 𝐼 ∈ V ) |
27 |
|
ovex |
⊢ ( 0 ... ( ♯ ‘ 𝑓 ) ) ∈ V |
28 |
1
|
fvexi |
⊢ 𝑉 ∈ V |
29 |
28
|
a1i |
⊢ ( ( 𝐺 ∈ 𝑊 ∧ 𝑓 ∈ Word dom 𝐼 ) → 𝑉 ∈ V ) |
30 |
|
mapex |
⊢ ( ( ( 0 ... ( ♯ ‘ 𝑓 ) ) ∈ V ∧ 𝑉 ∈ V ) → { 𝑝 ∣ 𝑝 : ( 0 ... ( ♯ ‘ 𝑓 ) ) ⟶ 𝑉 } ∈ V ) |
31 |
27 29 30
|
sylancr |
⊢ ( ( 𝐺 ∈ 𝑊 ∧ 𝑓 ∈ Word dom 𝐼 ) → { 𝑝 ∣ 𝑝 : ( 0 ... ( ♯ ‘ 𝑓 ) ) ⟶ 𝑉 } ∈ V ) |
32 |
|
simpl |
⊢ ( ( 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 ) ) ) ) → 𝑝 : ( 0 ... ( ♯ ‘ 𝑓 ) ) ⟶ 𝑉 ) |
33 |
32
|
ss2abi |
⊢ { 𝑝 ∣ ( 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 ) ) ) ) } ⊆ { 𝑝 ∣ 𝑝 : ( 0 ... ( ♯ ‘ 𝑓 ) ) ⟶ 𝑉 } |
34 |
33
|
a1i |
⊢ ( ( 𝐺 ∈ 𝑊 ∧ 𝑓 ∈ Word dom 𝐼 ) → { 𝑝 ∣ ( 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 ) ) ) ) } ⊆ { 𝑝 ∣ 𝑝 : ( 0 ... ( ♯ ‘ 𝑓 ) ) ⟶ 𝑉 } ) |
35 |
31 34
|
ssexd |
⊢ ( ( 𝐺 ∈ 𝑊 ∧ 𝑓 ∈ Word dom 𝐼 ) → { 𝑝 ∣ ( 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 ) ) ) ) } ∈ V ) |
36 |
26 35
|
opabex3d |
⊢ ( 𝐺 ∈ 𝑊 → { 〈 𝑓 , 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 ) ) ) ) ) } ∈ V ) |
37 |
22 36
|
eqeltrid |
⊢ ( 𝐺 ∈ 𝑊 → { 〈 𝑓 , 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 ) ) ) ) } ∈ V ) |
38 |
3 19 20 37
|
fvmptd3 |
⊢ ( 𝐺 ∈ 𝑊 → ( Walks ‘ 𝐺 ) = { 〈 𝑓 , 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 ) ) ) ) } ) |