Step |
Hyp |
Ref |
Expression |
1 |
|
wrdf |
|- ( W e. Word S -> W : ( 0 ..^ ( # ` W ) ) --> S ) |
2 |
|
elfzuz3 |
|- ( N e. ( 0 ... ( # ` W ) ) -> ( # ` W ) e. ( ZZ>= ` N ) ) |
3 |
|
fzoss2 |
|- ( ( # ` W ) e. ( ZZ>= ` N ) -> ( 0 ..^ N ) C_ ( 0 ..^ ( # ` W ) ) ) |
4 |
2 3
|
syl |
|- ( N e. ( 0 ... ( # ` W ) ) -> ( 0 ..^ N ) C_ ( 0 ..^ ( # ` W ) ) ) |
5 |
|
fssres |
|- ( ( W : ( 0 ..^ ( # ` W ) ) --> S /\ ( 0 ..^ N ) C_ ( 0 ..^ ( # ` W ) ) ) -> ( W |` ( 0 ..^ N ) ) : ( 0 ..^ N ) --> S ) |
6 |
1 4 5
|
syl2an |
|- ( ( W e. Word S /\ N e. ( 0 ... ( # ` W ) ) ) -> ( W |` ( 0 ..^ N ) ) : ( 0 ..^ N ) --> S ) |
7 |
|
iswrdi |
|- ( ( W |` ( 0 ..^ N ) ) : ( 0 ..^ N ) --> S -> ( W |` ( 0 ..^ N ) ) e. Word S ) |
8 |
6 7
|
syl |
|- ( ( W e. Word S /\ N e. ( 0 ... ( # ` W ) ) ) -> ( W |` ( 0 ..^ N ) ) e. Word S ) |