Step |
Hyp |
Ref |
Expression |
1 |
|
wlkson.v |
|- V = ( Vtx ` G ) |
2 |
1
|
fvexi |
|- V e. _V |
3 |
|
df-wlkson |
|- WalksOn = ( g e. _V |-> ( a e. ( Vtx ` g ) , b e. ( Vtx ` g ) |-> { <. f , p >. | ( f ( Walks ` g ) p /\ ( p ` 0 ) = a /\ ( p ` ( # ` f ) ) = b ) } ) ) |
4 |
1
|
wlkson |
|- ( ( A e. V /\ B e. V ) -> ( A ( WalksOn ` G ) B ) = { <. f , p >. | ( f ( Walks ` G ) p /\ ( p ` 0 ) = A /\ ( p ` ( # ` f ) ) = B ) } ) |
5 |
4
|
3adant1 |
|- ( ( G e. _V /\ A e. V /\ B e. V ) -> ( A ( WalksOn ` G ) B ) = { <. f , p >. | ( f ( Walks ` G ) p /\ ( p ` 0 ) = A /\ ( p ` ( # ` f ) ) = B ) } ) |
6 |
|
fveq2 |
|- ( g = G -> ( Vtx ` g ) = ( Vtx ` G ) ) |
7 |
6 1
|
eqtr4di |
|- ( g = G -> ( Vtx ` g ) = V ) |
8 |
|
fveq2 |
|- ( g = G -> ( Walks ` g ) = ( Walks ` G ) ) |
9 |
8
|
breqd |
|- ( g = G -> ( f ( Walks ` g ) p <-> f ( Walks ` G ) p ) ) |
10 |
9
|
3anbi1d |
|- ( g = G -> ( ( f ( Walks ` g ) p /\ ( p ` 0 ) = a /\ ( p ` ( # ` f ) ) = b ) <-> ( f ( Walks ` G ) p /\ ( p ` 0 ) = a /\ ( p ` ( # ` f ) ) = b ) ) ) |
11 |
3 5 7 7 10
|
bropfvvvv |
|- ( ( V e. _V /\ V e. _V ) -> ( F ( A ( WalksOn ` G ) B ) P -> ( G e. _V /\ ( A e. V /\ B e. V ) /\ ( F e. _V /\ P e. _V ) ) ) ) |
12 |
2 2 11
|
mp2an |
|- ( F ( A ( WalksOn ` G ) B ) P -> ( G e. _V /\ ( A e. V /\ B e. V ) /\ ( F e. _V /\ P e. _V ) ) ) |
13 |
|
3anass |
|- ( ( G e. _V /\ A e. V /\ B e. V ) <-> ( G e. _V /\ ( A e. V /\ B e. V ) ) ) |
14 |
13
|
anbi1i |
|- ( ( ( G e. _V /\ A e. V /\ B e. V ) /\ ( F e. _V /\ P e. _V ) ) <-> ( ( G e. _V /\ ( A e. V /\ B e. V ) ) /\ ( F e. _V /\ P e. _V ) ) ) |
15 |
|
df-3an |
|- ( ( G e. _V /\ ( A e. V /\ B e. V ) /\ ( F e. _V /\ P e. _V ) ) <-> ( ( G e. _V /\ ( A e. V /\ B e. V ) ) /\ ( F e. _V /\ P e. _V ) ) ) |
16 |
14 15
|
bitr4i |
|- ( ( ( G e. _V /\ A e. V /\ B e. V ) /\ ( F e. _V /\ P e. _V ) ) <-> ( G e. _V /\ ( A e. V /\ B e. V ) /\ ( F e. _V /\ P e. _V ) ) ) |
17 |
12 16
|
sylibr |
|- ( F ( A ( WalksOn ` G ) B ) P -> ( ( G e. _V /\ A e. V /\ B e. V ) /\ ( F e. _V /\ P e. _V ) ) ) |
18 |
1
|
iswlkon |
|- ( ( ( A e. V /\ B e. V ) /\ ( F e. _V /\ P e. _V ) ) -> ( F ( A ( WalksOn ` G ) B ) P <-> ( F ( Walks ` G ) P /\ ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) ) ) |
19 |
18
|
3adantl1 |
|- ( ( ( G e. _V /\ A e. V /\ B e. V ) /\ ( F e. _V /\ P e. _V ) ) -> ( F ( A ( WalksOn ` G ) B ) P <-> ( F ( Walks ` G ) P /\ ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) ) ) |
20 |
19
|
biimpd |
|- ( ( ( G e. _V /\ A e. V /\ B e. V ) /\ ( F e. _V /\ P e. _V ) ) -> ( F ( A ( WalksOn ` G ) B ) P -> ( F ( Walks ` G ) P /\ ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) ) ) |
21 |
20
|
imdistani |
|- ( ( ( ( G e. _V /\ A e. V /\ B e. V ) /\ ( F e. _V /\ P e. _V ) ) /\ F ( A ( WalksOn ` G ) B ) P ) -> ( ( ( G e. _V /\ A e. V /\ B e. V ) /\ ( F e. _V /\ P e. _V ) ) /\ ( F ( Walks ` G ) P /\ ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) ) ) |
22 |
17 21
|
mpancom |
|- ( F ( A ( WalksOn ` G ) B ) P -> ( ( ( G e. _V /\ A e. V /\ B e. V ) /\ ( F e. _V /\ P e. _V ) ) /\ ( F ( Walks ` G ) P /\ ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) ) ) |
23 |
|
df-3an |
|- ( ( ( G e. _V /\ A e. V /\ B e. V ) /\ ( F e. _V /\ P e. _V ) /\ ( F ( Walks ` G ) P /\ ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) ) <-> ( ( ( G e. _V /\ A e. V /\ B e. V ) /\ ( F e. _V /\ P e. _V ) ) /\ ( F ( Walks ` G ) P /\ ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) ) ) |
24 |
22 23
|
sylibr |
|- ( F ( A ( WalksOn ` G ) B ) P -> ( ( G e. _V /\ A e. V /\ B e. V ) /\ ( F e. _V /\ P e. _V ) /\ ( F ( Walks ` G ) P /\ ( P ` 0 ) = A /\ ( P ` ( # ` F ) ) = B ) ) ) |