| Step |
Hyp |
Ref |
Expression |
| 1 |
|
pfxf1.1 |
|- ( ph -> W e. Word S ) |
| 2 |
|
pfxf1.2 |
|- ( ph -> W : dom W -1-1-> S ) |
| 3 |
|
pfxf1.3 |
|- ( ph -> L e. ( 0 ... ( # ` W ) ) ) |
| 4 |
|
elfzuz3 |
|- ( L e. ( 0 ... ( # ` W ) ) -> ( # ` W ) e. ( ZZ>= ` L ) ) |
| 5 |
|
fzoss2 |
|- ( ( # ` W ) e. ( ZZ>= ` L ) -> ( 0 ..^ L ) C_ ( 0 ..^ ( # ` W ) ) ) |
| 6 |
3 4 5
|
3syl |
|- ( ph -> ( 0 ..^ L ) C_ ( 0 ..^ ( # ` W ) ) ) |
| 7 |
|
wrddm |
|- ( W e. Word S -> dom W = ( 0 ..^ ( # ` W ) ) ) |
| 8 |
1 7
|
syl |
|- ( ph -> dom W = ( 0 ..^ ( # ` W ) ) ) |
| 9 |
6 8
|
sseqtrrd |
|- ( ph -> ( 0 ..^ L ) C_ dom W ) |
| 10 |
|
wrdf |
|- ( W e. Word S -> W : ( 0 ..^ ( # ` W ) ) --> S ) |
| 11 |
1 10
|
syl |
|- ( ph -> W : ( 0 ..^ ( # ` W ) ) --> S ) |
| 12 |
11 6
|
fssresd |
|- ( ph -> ( W |` ( 0 ..^ L ) ) : ( 0 ..^ L ) --> S ) |
| 13 |
|
f1resf1 |
|- ( ( W : dom W -1-1-> S /\ ( 0 ..^ L ) C_ dom W /\ ( W |` ( 0 ..^ L ) ) : ( 0 ..^ L ) --> S ) -> ( W |` ( 0 ..^ L ) ) : ( 0 ..^ L ) -1-1-> S ) |
| 14 |
2 9 12 13
|
syl3anc |
|- ( ph -> ( W |` ( 0 ..^ L ) ) : ( 0 ..^ L ) -1-1-> S ) |
| 15 |
|
pfxres |
|- ( ( W e. Word S /\ L e. ( 0 ... ( # ` W ) ) ) -> ( W prefix L ) = ( W |` ( 0 ..^ L ) ) ) |
| 16 |
1 3 15
|
syl2anc |
|- ( ph -> ( W prefix L ) = ( W |` ( 0 ..^ L ) ) ) |
| 17 |
|
pfxfn |
|- ( ( W e. Word S /\ L e. ( 0 ... ( # ` W ) ) ) -> ( W prefix L ) Fn ( 0 ..^ L ) ) |
| 18 |
1 3 17
|
syl2anc |
|- ( ph -> ( W prefix L ) Fn ( 0 ..^ L ) ) |
| 19 |
18
|
fndmd |
|- ( ph -> dom ( W prefix L ) = ( 0 ..^ L ) ) |
| 20 |
|
eqidd |
|- ( ph -> S = S ) |
| 21 |
16 19 20
|
f1eq123d |
|- ( ph -> ( ( W prefix L ) : dom ( W prefix L ) -1-1-> S <-> ( W |` ( 0 ..^ L ) ) : ( 0 ..^ L ) -1-1-> S ) ) |
| 22 |
14 21
|
mpbird |
|- ( ph -> ( W prefix L ) : dom ( W prefix L ) -1-1-> S ) |