Step |
Hyp |
Ref |
Expression |
1 |
|
rusgrnumwwlk.v |
⊢ 𝑉 = ( Vtx ‘ 𝐺 ) |
2 |
|
rusgrnumwwlk.l |
⊢ 𝐿 = ( 𝑣 ∈ 𝑉 , 𝑛 ∈ ℕ0 ↦ ( ♯ ‘ { 𝑤 ∈ ( 𝑛 WWalksN 𝐺 ) ∣ ( 𝑤 ‘ 0 ) = 𝑣 } ) ) |
3 |
|
simpr |
⊢ ( ( 𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉 ) → 𝑃 ∈ 𝑉 ) |
4 |
|
0nn0 |
⊢ 0 ∈ ℕ0 |
5 |
1 2
|
rusgrnumwwlklem |
⊢ ( ( 𝑃 ∈ 𝑉 ∧ 0 ∈ ℕ0 ) → ( 𝑃 𝐿 0 ) = ( ♯ ‘ { 𝑤 ∈ ( 0 WWalksN 𝐺 ) ∣ ( 𝑤 ‘ 0 ) = 𝑃 } ) ) |
6 |
3 4 5
|
sylancl |
⊢ ( ( 𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉 ) → ( 𝑃 𝐿 0 ) = ( ♯ ‘ { 𝑤 ∈ ( 0 WWalksN 𝐺 ) ∣ ( 𝑤 ‘ 0 ) = 𝑃 } ) ) |
7 |
|
df-rab |
⊢ { 𝑤 ∈ ( 0 WWalksN 𝐺 ) ∣ ( 𝑤 ‘ 0 ) = 𝑃 } = { 𝑤 ∣ ( 𝑤 ∈ ( 0 WWalksN 𝐺 ) ∧ ( 𝑤 ‘ 0 ) = 𝑃 ) } |
8 |
7
|
a1i |
⊢ ( ( 𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉 ) → { 𝑤 ∈ ( 0 WWalksN 𝐺 ) ∣ ( 𝑤 ‘ 0 ) = 𝑃 } = { 𝑤 ∣ ( 𝑤 ∈ ( 0 WWalksN 𝐺 ) ∧ ( 𝑤 ‘ 0 ) = 𝑃 ) } ) |
9 |
|
wwlksn0s |
⊢ ( 0 WWalksN 𝐺 ) = { 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∣ ( ♯ ‘ 𝑤 ) = 1 } |
10 |
9
|
a1i |
⊢ ( ( 𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉 ) → ( 0 WWalksN 𝐺 ) = { 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∣ ( ♯ ‘ 𝑤 ) = 1 } ) |
11 |
10
|
eleq2d |
⊢ ( ( 𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉 ) → ( 𝑤 ∈ ( 0 WWalksN 𝐺 ) ↔ 𝑤 ∈ { 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∣ ( ♯ ‘ 𝑤 ) = 1 } ) ) |
12 |
|
rabid |
⊢ ( 𝑤 ∈ { 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∣ ( ♯ ‘ 𝑤 ) = 1 } ↔ ( 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑤 ) = 1 ) ) |
13 |
11 12
|
syl6bb |
⊢ ( ( 𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉 ) → ( 𝑤 ∈ ( 0 WWalksN 𝐺 ) ↔ ( 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑤 ) = 1 ) ) ) |
14 |
13
|
anbi1d |
⊢ ( ( 𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉 ) → ( ( 𝑤 ∈ ( 0 WWalksN 𝐺 ) ∧ ( 𝑤 ‘ 0 ) = 𝑃 ) ↔ ( ( 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑤 ) = 1 ) ∧ ( 𝑤 ‘ 0 ) = 𝑃 ) ) ) |
15 |
14
|
abbidv |
⊢ ( ( 𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉 ) → { 𝑤 ∣ ( 𝑤 ∈ ( 0 WWalksN 𝐺 ) ∧ ( 𝑤 ‘ 0 ) = 𝑃 ) } = { 𝑤 ∣ ( ( 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑤 ) = 1 ) ∧ ( 𝑤 ‘ 0 ) = 𝑃 ) } ) |
16 |
|
wrdl1s1 |
⊢ ( 𝑃 ∈ ( Vtx ‘ 𝐺 ) → ( 𝑣 = 〈“ 𝑃 ”〉 ↔ ( 𝑣 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑣 ) = 1 ∧ ( 𝑣 ‘ 0 ) = 𝑃 ) ) ) |
17 |
|
df-3an |
⊢ ( ( 𝑣 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑣 ) = 1 ∧ ( 𝑣 ‘ 0 ) = 𝑃 ) ↔ ( ( 𝑣 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑣 ) = 1 ) ∧ ( 𝑣 ‘ 0 ) = 𝑃 ) ) |
18 |
16 17
|
bitr2di |
⊢ ( 𝑃 ∈ ( Vtx ‘ 𝐺 ) → ( ( ( 𝑣 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑣 ) = 1 ) ∧ ( 𝑣 ‘ 0 ) = 𝑃 ) ↔ 𝑣 = 〈“ 𝑃 ”〉 ) ) |
19 |
|
vex |
⊢ 𝑣 ∈ V |
20 |
|
eleq1w |
⊢ ( 𝑤 = 𝑣 → ( 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ↔ 𝑣 ∈ Word ( Vtx ‘ 𝐺 ) ) ) |
21 |
|
fveqeq2 |
⊢ ( 𝑤 = 𝑣 → ( ( ♯ ‘ 𝑤 ) = 1 ↔ ( ♯ ‘ 𝑣 ) = 1 ) ) |
22 |
20 21
|
anbi12d |
⊢ ( 𝑤 = 𝑣 → ( ( 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑤 ) = 1 ) ↔ ( 𝑣 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑣 ) = 1 ) ) ) |
23 |
|
fveq1 |
⊢ ( 𝑤 = 𝑣 → ( 𝑤 ‘ 0 ) = ( 𝑣 ‘ 0 ) ) |
24 |
23
|
eqeq1d |
⊢ ( 𝑤 = 𝑣 → ( ( 𝑤 ‘ 0 ) = 𝑃 ↔ ( 𝑣 ‘ 0 ) = 𝑃 ) ) |
25 |
22 24
|
anbi12d |
⊢ ( 𝑤 = 𝑣 → ( ( ( 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑤 ) = 1 ) ∧ ( 𝑤 ‘ 0 ) = 𝑃 ) ↔ ( ( 𝑣 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑣 ) = 1 ) ∧ ( 𝑣 ‘ 0 ) = 𝑃 ) ) ) |
26 |
19 25
|
elab |
⊢ ( 𝑣 ∈ { 𝑤 ∣ ( ( 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑤 ) = 1 ) ∧ ( 𝑤 ‘ 0 ) = 𝑃 ) } ↔ ( ( 𝑣 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑣 ) = 1 ) ∧ ( 𝑣 ‘ 0 ) = 𝑃 ) ) |
27 |
|
velsn |
⊢ ( 𝑣 ∈ { 〈“ 𝑃 ”〉 } ↔ 𝑣 = 〈“ 𝑃 ”〉 ) |
28 |
18 26 27
|
3bitr4g |
⊢ ( 𝑃 ∈ ( Vtx ‘ 𝐺 ) → ( 𝑣 ∈ { 𝑤 ∣ ( ( 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑤 ) = 1 ) ∧ ( 𝑤 ‘ 0 ) = 𝑃 ) } ↔ 𝑣 ∈ { 〈“ 𝑃 ”〉 } ) ) |
29 |
28 1
|
eleq2s |
⊢ ( 𝑃 ∈ 𝑉 → ( 𝑣 ∈ { 𝑤 ∣ ( ( 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑤 ) = 1 ) ∧ ( 𝑤 ‘ 0 ) = 𝑃 ) } ↔ 𝑣 ∈ { 〈“ 𝑃 ”〉 } ) ) |
30 |
29
|
eqrdv |
⊢ ( 𝑃 ∈ 𝑉 → { 𝑤 ∣ ( ( 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑤 ) = 1 ) ∧ ( 𝑤 ‘ 0 ) = 𝑃 ) } = { 〈“ 𝑃 ”〉 } ) |
31 |
30
|
adantl |
⊢ ( ( 𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉 ) → { 𝑤 ∣ ( ( 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∧ ( ♯ ‘ 𝑤 ) = 1 ) ∧ ( 𝑤 ‘ 0 ) = 𝑃 ) } = { 〈“ 𝑃 ”〉 } ) |
32 |
8 15 31
|
3eqtrd |
⊢ ( ( 𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉 ) → { 𝑤 ∈ ( 0 WWalksN 𝐺 ) ∣ ( 𝑤 ‘ 0 ) = 𝑃 } = { 〈“ 𝑃 ”〉 } ) |
33 |
32
|
fveq2d |
⊢ ( ( 𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉 ) → ( ♯ ‘ { 𝑤 ∈ ( 0 WWalksN 𝐺 ) ∣ ( 𝑤 ‘ 0 ) = 𝑃 } ) = ( ♯ ‘ { 〈“ 𝑃 ”〉 } ) ) |
34 |
|
s1cl |
⊢ ( 𝑃 ∈ 𝑉 → 〈“ 𝑃 ”〉 ∈ Word 𝑉 ) |
35 |
|
hashsng |
⊢ ( 〈“ 𝑃 ”〉 ∈ Word 𝑉 → ( ♯ ‘ { 〈“ 𝑃 ”〉 } ) = 1 ) |
36 |
34 35
|
syl |
⊢ ( 𝑃 ∈ 𝑉 → ( ♯ ‘ { 〈“ 𝑃 ”〉 } ) = 1 ) |
37 |
36
|
adantl |
⊢ ( ( 𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉 ) → ( ♯ ‘ { 〈“ 𝑃 ”〉 } ) = 1 ) |
38 |
6 33 37
|
3eqtrd |
⊢ ( ( 𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉 ) → ( 𝑃 𝐿 0 ) = 1 ) |