Step |
Hyp |
Ref |
Expression |
1 |
|
wlkson.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 |
|
eqeq2 |
|- ( a = A -> ( ( p ` 0 ) = a <-> ( p ` 0 ) = A ) ) |
9 |
|
eqeq2 |
|- ( b = B -> ( ( p ` ( # ` f ) ) = b <-> ( p ` ( # ` f ) ) = B ) ) |
10 |
8 9
|
bi2anan9 |
|- ( ( a = A /\ b = B ) -> ( ( ( p ` 0 ) = a /\ ( p ` ( # ` f ) ) = b ) <-> ( ( p ` 0 ) = A /\ ( p ` ( # ` f ) ) = B ) ) ) |
11 |
|
biidd |
|- ( g = G -> ( ( ( p ` 0 ) = a /\ ( p ` ( # ` f ) ) = b ) <-> ( ( p ` 0 ) = a /\ ( p ` ( # ` f ) ) = b ) ) ) |
12 |
|
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 ) } ) ) |
13 |
|
eqid |
|- ( Vtx ` g ) = ( Vtx ` g ) |
14 |
|
3anass |
|- ( ( f ( Walks ` g ) p /\ ( p ` 0 ) = a /\ ( p ` ( # ` f ) ) = b ) <-> ( f ( Walks ` g ) p /\ ( ( p ` 0 ) = a /\ ( p ` ( # ` f ) ) = b ) ) ) |
15 |
14
|
biancomi |
|- ( ( f ( Walks ` g ) p /\ ( p ` 0 ) = a /\ ( p ` ( # ` f ) ) = b ) <-> ( ( ( p ` 0 ) = a /\ ( p ` ( # ` f ) ) = b ) /\ f ( Walks ` g ) p ) ) |
16 |
15
|
opabbii |
|- { <. f , p >. | ( f ( Walks ` g ) p /\ ( p ` 0 ) = a /\ ( p ` ( # ` f ) ) = b ) } = { <. f , p >. | ( ( ( p ` 0 ) = a /\ ( p ` ( # ` f ) ) = b ) /\ f ( Walks ` g ) p ) } |
17 |
13 13 16
|
mpoeq123i |
|- ( a e. ( Vtx ` g ) , b e. ( Vtx ` g ) |-> { <. f , p >. | ( f ( Walks ` g ) p /\ ( p ` 0 ) = a /\ ( p ` ( # ` f ) ) = b ) } ) = ( a e. ( Vtx ` g ) , b e. ( Vtx ` g ) |-> { <. f , p >. | ( ( ( p ` 0 ) = a /\ ( p ` ( # ` f ) ) = b ) /\ f ( Walks ` g ) p ) } ) |
18 |
17
|
mpteq2i |
|- ( g e. _V |-> ( a e. ( Vtx ` g ) , b e. ( Vtx ` g ) |-> { <. f , p >. | ( f ( Walks ` g ) p /\ ( p ` 0 ) = a /\ ( p ` ( # ` f ) ) = b ) } ) ) = ( g e. _V |-> ( a e. ( Vtx ` g ) , b e. ( Vtx ` g ) |-> { <. f , p >. | ( ( ( p ` 0 ) = a /\ ( p ` ( # ` f ) ) = b ) /\ f ( Walks ` g ) p ) } ) ) |
19 |
12 18
|
eqtri |
|- WalksOn = ( g e. _V |-> ( a e. ( Vtx ` g ) , b e. ( Vtx ` g ) |-> { <. f , p >. | ( ( ( p ` 0 ) = a /\ ( p ` ( # ` f ) ) = b ) /\ f ( Walks ` g ) p ) } ) ) |
20 |
3 5 7 10 11 19
|
mptmpoopabbrd |
|- ( ( A e. V /\ B e. V ) -> ( A ( WalksOn ` G ) B ) = { <. f , p >. | ( ( ( p ` 0 ) = A /\ ( p ` ( # ` f ) ) = B ) /\ f ( Walks ` G ) p ) } ) |
21 |
|
ancom |
|- ( ( ( ( p ` 0 ) = A /\ ( p ` ( # ` f ) ) = B ) /\ f ( Walks ` G ) p ) <-> ( f ( Walks ` G ) p /\ ( ( p ` 0 ) = A /\ ( p ` ( # ` f ) ) = B ) ) ) |
22 |
|
3anass |
|- ( ( f ( Walks ` G ) p /\ ( p ` 0 ) = A /\ ( p ` ( # ` f ) ) = B ) <-> ( f ( Walks ` G ) p /\ ( ( p ` 0 ) = A /\ ( p ` ( # ` f ) ) = B ) ) ) |
23 |
21 22
|
bitr4i |
|- ( ( ( ( p ` 0 ) = A /\ ( p ` ( # ` f ) ) = B ) /\ f ( Walks ` G ) p ) <-> ( f ( Walks ` G ) p /\ ( p ` 0 ) = A /\ ( p ` ( # ` f ) ) = B ) ) |
24 |
23
|
opabbii |
|- { <. f , p >. | ( ( ( p ` 0 ) = A /\ ( p ` ( # ` f ) ) = B ) /\ f ( Walks ` G ) p ) } = { <. f , p >. | ( f ( Walks ` G ) p /\ ( p ` 0 ) = A /\ ( p ` ( # ` f ) ) = B ) } |
25 |
20 24
|
eqtrdi |
|- ( ( A e. V /\ B e. V ) -> ( A ( WalksOn ` G ) B ) = { <. f , p >. | ( f ( Walks ` G ) p /\ ( p ` 0 ) = A /\ ( p ` ( # ` f ) ) = B ) } ) |