Step |
Hyp |
Ref |
Expression |
1 |
|
0wlk.v |
|- V = ( Vtx ` G ) |
2 |
|
s1val |
|- ( N e. V -> <" N "> = { <. 0 , N >. } ) |
3 |
|
0z |
|- 0 e. ZZ |
4 |
3
|
jctl |
|- ( N e. V -> ( 0 e. ZZ /\ N e. V ) ) |
5 |
|
f1sng |
|- ( ( 0 e. ZZ /\ N e. V ) -> { <. 0 , N >. } : { 0 } -1-1-> V ) |
6 |
|
f1f |
|- ( { <. 0 , N >. } : { 0 } -1-1-> V -> { <. 0 , N >. } : { 0 } --> V ) |
7 |
4 5 6
|
3syl |
|- ( N e. V -> { <. 0 , N >. } : { 0 } --> V ) |
8 |
|
id |
|- ( <" N "> = { <. 0 , N >. } -> <" N "> = { <. 0 , N >. } ) |
9 |
|
fzsn |
|- ( 0 e. ZZ -> ( 0 ... 0 ) = { 0 } ) |
10 |
3 9
|
mp1i |
|- ( <" N "> = { <. 0 , N >. } -> ( 0 ... 0 ) = { 0 } ) |
11 |
8 10
|
feq12d |
|- ( <" N "> = { <. 0 , N >. } -> ( <" N "> : ( 0 ... 0 ) --> V <-> { <. 0 , N >. } : { 0 } --> V ) ) |
12 |
7 11
|
syl5ibrcom |
|- ( N e. V -> ( <" N "> = { <. 0 , N >. } -> <" N "> : ( 0 ... 0 ) --> V ) ) |
13 |
2 12
|
mpd |
|- ( N e. V -> <" N "> : ( 0 ... 0 ) --> V ) |
14 |
|
s1fv |
|- ( N e. V -> ( <" N "> ` 0 ) = N ) |
15 |
1
|
0wlkon |
|- ( ( <" N "> : ( 0 ... 0 ) --> V /\ ( <" N "> ` 0 ) = N ) -> (/) ( N ( WalksOn ` G ) N ) <" N "> ) |
16 |
13 14 15
|
syl2anc |
|- ( N e. V -> (/) ( N ( WalksOn ` G ) N ) <" N "> ) |