Step |
Hyp |
Ref |
Expression |
1 |
|
iedginwlk.i |
|- I = ( iEdg ` G ) |
2 |
|
simp1 |
|- ( ( Fun I /\ F ( Walks ` G ) P /\ X e. ( 0 ..^ ( # ` F ) ) ) -> Fun I ) |
3 |
1
|
wlkf |
|- ( F ( Walks ` G ) P -> F e. Word dom I ) |
4 |
3
|
3ad2ant2 |
|- ( ( Fun I /\ F ( Walks ` G ) P /\ X e. ( 0 ..^ ( # ` F ) ) ) -> F e. Word dom I ) |
5 |
|
simp3 |
|- ( ( Fun I /\ F ( Walks ` G ) P /\ X e. ( 0 ..^ ( # ` F ) ) ) -> X e. ( 0 ..^ ( # ` F ) ) ) |
6 |
|
wrdsymbcl |
|- ( ( F e. Word dom I /\ X e. ( 0 ..^ ( # ` F ) ) ) -> ( F ` X ) e. dom I ) |
7 |
4 5 6
|
syl2anc |
|- ( ( Fun I /\ F ( Walks ` G ) P /\ X e. ( 0 ..^ ( # ` F ) ) ) -> ( F ` X ) e. dom I ) |
8 |
|
fvelrn |
|- ( ( Fun I /\ ( F ` X ) e. dom I ) -> ( I ` ( F ` X ) ) e. ran I ) |
9 |
2 7 8
|
syl2anc |
|- ( ( Fun I /\ F ( Walks ` G ) P /\ X e. ( 0 ..^ ( # ` F ) ) ) -> ( I ` ( F ` X ) ) e. ran I ) |