Step |
Hyp |
Ref |
Expression |
1 |
|
pthsonfval.v |
|- V = ( Vtx ` G ) |
2 |
1
|
1vgrex |
|- ( A e. V -> G e. _V ) |
3 |
2
|
adantr |
|- ( ( A e. V /\ B e. V ) -> G e. _V ) |
4 |
|
simpl |
|- ( ( A e. V /\ B e. V ) -> A e. V ) |
5 |
4 1
|
eleqtrdi |
|- ( ( A e. V /\ B e. V ) -> A e. ( Vtx ` G ) ) |
6 |
|
simpr |
|- ( ( A e. V /\ B e. V ) -> B e. V ) |
7 |
6 1
|
eleqtrdi |
|- ( ( A e. V /\ B e. V ) -> B e. ( Vtx ` G ) ) |
8 |
|
wksv |
|- { <. f , p >. | f ( Walks ` G ) p } e. _V |
9 |
8
|
a1i |
|- ( ( A e. V /\ B e. V ) -> { <. f , p >. | f ( Walks ` G ) p } e. _V ) |
10 |
|
pthiswlk |
|- ( f ( Paths ` G ) p -> f ( Walks ` G ) p ) |
11 |
10
|
adantl |
|- ( ( ( A e. V /\ B e. V ) /\ f ( Paths ` G ) p ) -> f ( Walks ` G ) p ) |
12 |
|
df-pthson |
|- PathsOn = ( g e. _V |-> ( a e. ( Vtx ` g ) , b e. ( Vtx ` g ) |-> { <. f , p >. | ( f ( a ( TrailsOn ` g ) b ) p /\ f ( Paths ` g ) p ) } ) ) |
13 |
3 5 7 9 11 12
|
mptmpoopabovd |
|- ( ( A e. V /\ B e. V ) -> ( A ( PathsOn ` G ) B ) = { <. f , p >. | ( f ( A ( TrailsOn ` G ) B ) p /\ f ( Paths ` G ) p ) } ) |