Step |
Hyp |
Ref |
Expression |
1 |
|
wlkvtxeledg.i |
|- I = ( iEdg ` G ) |
2 |
1
|
wlkvtxeledg |
|- ( F ( Walks ` G ) P -> A. k e. ( 0 ..^ ( # ` F ) ) { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) |
3 |
|
fvex |
|- ( P ` k ) e. _V |
4 |
3
|
prnz |
|- { ( P ` k ) , ( P ` ( k + 1 ) ) } =/= (/) |
5 |
|
ssn0 |
|- ( ( { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) /\ { ( P ` k ) , ( P ` ( k + 1 ) ) } =/= (/) ) -> ( I ` ( F ` k ) ) =/= (/) ) |
6 |
4 5
|
mpan2 |
|- ( { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) -> ( I ` ( F ` k ) ) =/= (/) ) |
7 |
6
|
adantl |
|- ( ( ( F ( Walks ` G ) P /\ k e. ( 0 ..^ ( # ` F ) ) ) /\ { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) -> ( I ` ( F ` k ) ) =/= (/) ) |
8 |
|
fvn0fvelrn |
|- ( ( I ` ( F ` k ) ) =/= (/) -> ( I ` ( F ` k ) ) e. ran I ) |
9 |
7 8
|
syl |
|- ( ( ( F ( Walks ` G ) P /\ k e. ( 0 ..^ ( # ` F ) ) ) /\ { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) -> ( I ` ( F ` k ) ) e. ran I ) |
10 |
|
sseq2 |
|- ( e = ( I ` ( F ` k ) ) -> ( { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ e <-> { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) ) |
11 |
10
|
adantl |
|- ( ( ( ( F ( Walks ` G ) P /\ k e. ( 0 ..^ ( # ` F ) ) ) /\ { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) /\ e = ( I ` ( F ` k ) ) ) -> ( { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ e <-> { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) ) |
12 |
|
simpr |
|- ( ( ( F ( Walks ` G ) P /\ k e. ( 0 ..^ ( # ` F ) ) ) /\ { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) -> { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) |
13 |
9 11 12
|
rspcedvd |
|- ( ( ( F ( Walks ` G ) P /\ k e. ( 0 ..^ ( # ` F ) ) ) /\ { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) -> E. e e. ran I { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ e ) |
14 |
13
|
ex |
|- ( ( F ( Walks ` G ) P /\ k e. ( 0 ..^ ( # ` F ) ) ) -> ( { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) -> E. e e. ran I { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ e ) ) |
15 |
14
|
ralimdva |
|- ( F ( Walks ` G ) P -> ( A. k e. ( 0 ..^ ( # ` F ) ) { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) -> A. k e. ( 0 ..^ ( # ` F ) ) E. e e. ran I { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ e ) ) |
16 |
2 15
|
mpd |
|- ( F ( Walks ` G ) P -> A. k e. ( 0 ..^ ( # ` F ) ) E. e e. ran I { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ e ) |