Step |
Hyp |
Ref |
Expression |
1 |
|
wrdnfi |
⊢ ( ( Vtx ‘ 𝐺 ) ∈ Fin → { 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∣ ( ♯ ‘ 𝑤 ) = ( 𝑁 + 1 ) } ∈ Fin ) |
2 |
|
simpr |
⊢ ( ( ( 𝑤 ≠ ∅ ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑤 ) − 1 ) ) { ( 𝑤 ‘ 𝑖 ) , ( 𝑤 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ) ∧ ( ♯ ‘ 𝑤 ) = ( 𝑁 + 1 ) ) → ( ♯ ‘ 𝑤 ) = ( 𝑁 + 1 ) ) |
3 |
2
|
a1i |
⊢ ( 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) → ( ( ( 𝑤 ≠ ∅ ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑤 ) − 1 ) ) { ( 𝑤 ‘ 𝑖 ) , ( 𝑤 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ) ∧ ( ♯ ‘ 𝑤 ) = ( 𝑁 + 1 ) ) → ( ♯ ‘ 𝑤 ) = ( 𝑁 + 1 ) ) ) |
4 |
3
|
ss2rabi |
⊢ { 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∣ ( ( 𝑤 ≠ ∅ ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑤 ) − 1 ) ) { ( 𝑤 ‘ 𝑖 ) , ( 𝑤 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ) ∧ ( ♯ ‘ 𝑤 ) = ( 𝑁 + 1 ) ) } ⊆ { 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∣ ( ♯ ‘ 𝑤 ) = ( 𝑁 + 1 ) } |
5 |
4
|
a1i |
⊢ ( ( Vtx ‘ 𝐺 ) ∈ Fin → { 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∣ ( ( 𝑤 ≠ ∅ ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑤 ) − 1 ) ) { ( 𝑤 ‘ 𝑖 ) , ( 𝑤 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ) ∧ ( ♯ ‘ 𝑤 ) = ( 𝑁 + 1 ) ) } ⊆ { 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∣ ( ♯ ‘ 𝑤 ) = ( 𝑁 + 1 ) } ) |
6 |
1 5
|
ssfid |
⊢ ( ( Vtx ‘ 𝐺 ) ∈ Fin → { 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∣ ( ( 𝑤 ≠ ∅ ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑤 ) − 1 ) ) { ( 𝑤 ‘ 𝑖 ) , ( 𝑤 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ) ∧ ( ♯ ‘ 𝑤 ) = ( 𝑁 + 1 ) ) } ∈ Fin ) |
7 |
|
wwlksn |
⊢ ( 𝑁 ∈ ℕ0 → ( 𝑁 WWalksN 𝐺 ) = { 𝑤 ∈ ( WWalks ‘ 𝐺 ) ∣ ( ♯ ‘ 𝑤 ) = ( 𝑁 + 1 ) } ) |
8 |
|
df-rab |
⊢ { 𝑤 ∈ ( WWalks ‘ 𝐺 ) ∣ ( ♯ ‘ 𝑤 ) = ( 𝑁 + 1 ) } = { 𝑤 ∣ ( 𝑤 ∈ ( WWalks ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑤 ) = ( 𝑁 + 1 ) ) } |
9 |
7 8
|
eqtrdi |
⊢ ( 𝑁 ∈ ℕ0 → ( 𝑁 WWalksN 𝐺 ) = { 𝑤 ∣ ( 𝑤 ∈ ( WWalks ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑤 ) = ( 𝑁 + 1 ) ) } ) |
10 |
|
3anan12 |
⊢ ( ( 𝑤 ≠ ∅ ∧ 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑤 ) − 1 ) ) { ( 𝑤 ‘ 𝑖 ) , ( 𝑤 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ) ↔ ( 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( 𝑤 ≠ ∅ ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑤 ) − 1 ) ) { ( 𝑤 ‘ 𝑖 ) , ( 𝑤 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ) ) ) |
11 |
10
|
anbi1i |
⊢ ( ( ( 𝑤 ≠ ∅ ∧ 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑤 ) − 1 ) ) { ( 𝑤 ‘ 𝑖 ) , ( 𝑤 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ) ∧ ( ♯ ‘ 𝑤 ) = ( 𝑁 + 1 ) ) ↔ ( ( 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( 𝑤 ≠ ∅ ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑤 ) − 1 ) ) { ( 𝑤 ‘ 𝑖 ) , ( 𝑤 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ) ) ∧ ( ♯ ‘ 𝑤 ) = ( 𝑁 + 1 ) ) ) |
12 |
|
anass |
⊢ ( ( ( 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( 𝑤 ≠ ∅ ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑤 ) − 1 ) ) { ( 𝑤 ‘ 𝑖 ) , ( 𝑤 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ) ) ∧ ( ♯ ‘ 𝑤 ) = ( 𝑁 + 1 ) ) ↔ ( 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ( 𝑤 ≠ ∅ ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑤 ) − 1 ) ) { ( 𝑤 ‘ 𝑖 ) , ( 𝑤 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ) ∧ ( ♯ ‘ 𝑤 ) = ( 𝑁 + 1 ) ) ) ) |
13 |
11 12
|
bitri |
⊢ ( ( ( 𝑤 ≠ ∅ ∧ 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑤 ) − 1 ) ) { ( 𝑤 ‘ 𝑖 ) , ( 𝑤 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ) ∧ ( ♯ ‘ 𝑤 ) = ( 𝑁 + 1 ) ) ↔ ( 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ( 𝑤 ≠ ∅ ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑤 ) − 1 ) ) { ( 𝑤 ‘ 𝑖 ) , ( 𝑤 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ) ∧ ( ♯ ‘ 𝑤 ) = ( 𝑁 + 1 ) ) ) ) |
14 |
13
|
abbii |
⊢ { 𝑤 ∣ ( ( 𝑤 ≠ ∅ ∧ 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑤 ) − 1 ) ) { ( 𝑤 ‘ 𝑖 ) , ( 𝑤 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ) ∧ ( ♯ ‘ 𝑤 ) = ( 𝑁 + 1 ) ) } = { 𝑤 ∣ ( 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ( 𝑤 ≠ ∅ ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑤 ) − 1 ) ) { ( 𝑤 ‘ 𝑖 ) , ( 𝑤 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ) ∧ ( ♯ ‘ 𝑤 ) = ( 𝑁 + 1 ) ) ) } |
15 |
|
eqid |
⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 ) |
16 |
|
eqid |
⊢ ( Edg ‘ 𝐺 ) = ( Edg ‘ 𝐺 ) |
17 |
15 16
|
iswwlks |
⊢ ( 𝑤 ∈ ( WWalks ‘ 𝐺 ) ↔ ( 𝑤 ≠ ∅ ∧ 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑤 ) − 1 ) ) { ( 𝑤 ‘ 𝑖 ) , ( 𝑤 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ) ) |
18 |
17
|
anbi1i |
⊢ ( ( 𝑤 ∈ ( WWalks ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑤 ) = ( 𝑁 + 1 ) ) ↔ ( ( 𝑤 ≠ ∅ ∧ 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑤 ) − 1 ) ) { ( 𝑤 ‘ 𝑖 ) , ( 𝑤 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ) ∧ ( ♯ ‘ 𝑤 ) = ( 𝑁 + 1 ) ) ) |
19 |
18
|
abbii |
⊢ { 𝑤 ∣ ( 𝑤 ∈ ( WWalks ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑤 ) = ( 𝑁 + 1 ) ) } = { 𝑤 ∣ ( ( 𝑤 ≠ ∅ ∧ 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑤 ) − 1 ) ) { ( 𝑤 ‘ 𝑖 ) , ( 𝑤 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ) ∧ ( ♯ ‘ 𝑤 ) = ( 𝑁 + 1 ) ) } |
20 |
|
df-rab |
⊢ { 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∣ ( ( 𝑤 ≠ ∅ ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑤 ) − 1 ) ) { ( 𝑤 ‘ 𝑖 ) , ( 𝑤 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ) ∧ ( ♯ ‘ 𝑤 ) = ( 𝑁 + 1 ) ) } = { 𝑤 ∣ ( 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ( 𝑤 ≠ ∅ ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑤 ) − 1 ) ) { ( 𝑤 ‘ 𝑖 ) , ( 𝑤 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ) ∧ ( ♯ ‘ 𝑤 ) = ( 𝑁 + 1 ) ) ) } |
21 |
14 19 20
|
3eqtr4i |
⊢ { 𝑤 ∣ ( 𝑤 ∈ ( WWalks ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑤 ) = ( 𝑁 + 1 ) ) } = { 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∣ ( ( 𝑤 ≠ ∅ ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑤 ) − 1 ) ) { ( 𝑤 ‘ 𝑖 ) , ( 𝑤 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ) ∧ ( ♯ ‘ 𝑤 ) = ( 𝑁 + 1 ) ) } |
22 |
9 21
|
eqtrdi |
⊢ ( 𝑁 ∈ ℕ0 → ( 𝑁 WWalksN 𝐺 ) = { 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∣ ( ( 𝑤 ≠ ∅ ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑤 ) − 1 ) ) { ( 𝑤 ‘ 𝑖 ) , ( 𝑤 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ) ∧ ( ♯ ‘ 𝑤 ) = ( 𝑁 + 1 ) ) } ) |
23 |
22
|
eleq1d |
⊢ ( 𝑁 ∈ ℕ0 → ( ( 𝑁 WWalksN 𝐺 ) ∈ Fin ↔ { 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∣ ( ( 𝑤 ≠ ∅ ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ( ♯ ‘ 𝑤 ) − 1 ) ) { ( 𝑤 ‘ 𝑖 ) , ( 𝑤 ‘ ( 𝑖 + 1 ) ) } ∈ ( Edg ‘ 𝐺 ) ) ∧ ( ♯ ‘ 𝑤 ) = ( 𝑁 + 1 ) ) } ∈ Fin ) ) |
24 |
6 23
|
imbitrrid |
⊢ ( 𝑁 ∈ ℕ0 → ( ( Vtx ‘ 𝐺 ) ∈ Fin → ( 𝑁 WWalksN 𝐺 ) ∈ Fin ) ) |
25 |
|
df-nel |
⊢ ( 𝑁 ∉ ℕ0 ↔ ¬ 𝑁 ∈ ℕ0 ) |
26 |
25
|
biimpri |
⊢ ( ¬ 𝑁 ∈ ℕ0 → 𝑁 ∉ ℕ0 ) |
27 |
26
|
olcd |
⊢ ( ¬ 𝑁 ∈ ℕ0 → ( 𝐺 ∉ V ∨ 𝑁 ∉ ℕ0 ) ) |
28 |
|
wwlksnndef |
⊢ ( ( 𝐺 ∉ V ∨ 𝑁 ∉ ℕ0 ) → ( 𝑁 WWalksN 𝐺 ) = ∅ ) |
29 |
27 28
|
syl |
⊢ ( ¬ 𝑁 ∈ ℕ0 → ( 𝑁 WWalksN 𝐺 ) = ∅ ) |
30 |
|
0fin |
⊢ ∅ ∈ Fin |
31 |
29 30
|
eqeltrdi |
⊢ ( ¬ 𝑁 ∈ ℕ0 → ( 𝑁 WWalksN 𝐺 ) ∈ Fin ) |
32 |
31
|
a1d |
⊢ ( ¬ 𝑁 ∈ ℕ0 → ( ( Vtx ‘ 𝐺 ) ∈ Fin → ( 𝑁 WWalksN 𝐺 ) ∈ Fin ) ) |
33 |
24 32
|
pm2.61i |
⊢ ( ( Vtx ‘ 𝐺 ) ∈ Fin → ( 𝑁 WWalksN 𝐺 ) ∈ Fin ) |