Step |
Hyp |
Ref |
Expression |
0 |
|
ctrlson |
|- TrailsOn |
1 |
|
vg |
|- g |
2 |
|
cvv |
|- _V |
3 |
|
va |
|- a |
4 |
|
cvtx |
|- Vtx |
5 |
1
|
cv |
|- g |
6 |
5 4
|
cfv |
|- ( Vtx ` g ) |
7 |
|
vb |
|- b |
8 |
|
vf |
|- f |
9 |
|
vp |
|- p |
10 |
8
|
cv |
|- f |
11 |
3
|
cv |
|- a |
12 |
|
cwlkson |
|- WalksOn |
13 |
5 12
|
cfv |
|- ( WalksOn ` g ) |
14 |
7
|
cv |
|- b |
15 |
11 14 13
|
co |
|- ( a ( WalksOn ` g ) b ) |
16 |
9
|
cv |
|- p |
17 |
10 16 15
|
wbr |
|- f ( a ( WalksOn ` g ) b ) p |
18 |
|
ctrls |
|- Trails |
19 |
5 18
|
cfv |
|- ( Trails ` g ) |
20 |
10 16 19
|
wbr |
|- f ( Trails ` g ) p |
21 |
17 20
|
wa |
|- ( f ( a ( WalksOn ` g ) b ) p /\ f ( Trails ` g ) p ) |
22 |
21 8 9
|
copab |
|- { <. f , p >. | ( f ( a ( WalksOn ` g ) b ) p /\ f ( Trails ` g ) p ) } |
23 |
3 7 6 6 22
|
cmpo |
|- ( a e. ( Vtx ` g ) , b e. ( Vtx ` g ) |-> { <. f , p >. | ( f ( a ( WalksOn ` g ) b ) p /\ f ( Trails ` g ) p ) } ) |
24 |
1 2 23
|
cmpt |
|- ( g e. _V |-> ( a e. ( Vtx ` g ) , b e. ( Vtx ` g ) |-> { <. f , p >. | ( f ( a ( WalksOn ` g ) b ) p /\ f ( Trails ` g ) p ) } ) ) |
25 |
0 24
|
wceq |
|- TrailsOn = ( g e. _V |-> ( a e. ( Vtx ` g ) , b e. ( Vtx ` g ) |-> { <. f , p >. | ( f ( a ( WalksOn ` g ) b ) p /\ f ( Trails ` g ) p ) } ) ) |