Step |
Hyp |
Ref |
Expression |
1 |
|
wlkop |
|- ( W e. ( Walks ` G ) -> W = <. ( 1st ` W ) , ( 2nd ` W ) >. ) |
2 |
|
wlkvv |
|- ( ( 1st ` W ) ( Walks ` G ) ( 2nd ` W ) -> W e. ( _V X. _V ) ) |
3 |
|
1st2ndb |
|- ( W e. ( _V X. _V ) <-> W = <. ( 1st ` W ) , ( 2nd ` W ) >. ) |
4 |
2 3
|
sylib |
|- ( ( 1st ` W ) ( Walks ` G ) ( 2nd ` W ) -> W = <. ( 1st ` W ) , ( 2nd ` W ) >. ) |
5 |
|
eleq1 |
|- ( W = <. ( 1st ` W ) , ( 2nd ` W ) >. -> ( W e. ( Walks ` G ) <-> <. ( 1st ` W ) , ( 2nd ` W ) >. e. ( Walks ` G ) ) ) |
6 |
|
df-br |
|- ( ( 1st ` W ) ( Walks ` G ) ( 2nd ` W ) <-> <. ( 1st ` W ) , ( 2nd ` W ) >. e. ( Walks ` G ) ) |
7 |
5 6
|
bitr4di |
|- ( W = <. ( 1st ` W ) , ( 2nd ` W ) >. -> ( W e. ( Walks ` G ) <-> ( 1st ` W ) ( Walks ` G ) ( 2nd ` W ) ) ) |
8 |
1 4 7
|
pm5.21nii |
|- ( W e. ( Walks ` G ) <-> ( 1st ` W ) ( Walks ` G ) ( 2nd ` W ) ) |