Step |
Hyp |
Ref |
Expression |
1 |
|
s3rn.i |
|- ( ph -> I e. D ) |
2 |
|
s3rn.j |
|- ( ph -> J e. D ) |
3 |
|
s3rn.k |
|- ( ph -> K e. D ) |
4 |
|
imadmrn |
|- ( <" I J K "> " dom <" I J K "> ) = ran <" I J K "> |
5 |
1 2 3
|
s3cld |
|- ( ph -> <" I J K "> e. Word D ) |
6 |
|
wrdfn |
|- ( <" I J K "> e. Word D -> <" I J K "> Fn ( 0 ..^ ( # ` <" I J K "> ) ) ) |
7 |
|
s3len |
|- ( # ` <" I J K "> ) = 3 |
8 |
7
|
oveq2i |
|- ( 0 ..^ ( # ` <" I J K "> ) ) = ( 0 ..^ 3 ) |
9 |
|
fzo0to3tp |
|- ( 0 ..^ 3 ) = { 0 , 1 , 2 } |
10 |
8 9
|
eqtri |
|- ( 0 ..^ ( # ` <" I J K "> ) ) = { 0 , 1 , 2 } |
11 |
10
|
fneq2i |
|- ( <" I J K "> Fn ( 0 ..^ ( # ` <" I J K "> ) ) <-> <" I J K "> Fn { 0 , 1 , 2 } ) |
12 |
11
|
biimpi |
|- ( <" I J K "> Fn ( 0 ..^ ( # ` <" I J K "> ) ) -> <" I J K "> Fn { 0 , 1 , 2 } ) |
13 |
5 6 12
|
3syl |
|- ( ph -> <" I J K "> Fn { 0 , 1 , 2 } ) |
14 |
13
|
fndmd |
|- ( ph -> dom <" I J K "> = { 0 , 1 , 2 } ) |
15 |
14
|
imaeq2d |
|- ( ph -> ( <" I J K "> " dom <" I J K "> ) = ( <" I J K "> " { 0 , 1 , 2 } ) ) |
16 |
|
c0ex |
|- 0 e. _V |
17 |
16
|
tpid1 |
|- 0 e. { 0 , 1 , 2 } |
18 |
17
|
a1i |
|- ( ph -> 0 e. { 0 , 1 , 2 } ) |
19 |
|
1ex |
|- 1 e. _V |
20 |
19
|
tpid2 |
|- 1 e. { 0 , 1 , 2 } |
21 |
20
|
a1i |
|- ( ph -> 1 e. { 0 , 1 , 2 } ) |
22 |
|
2ex |
|- 2 e. _V |
23 |
22
|
tpid3 |
|- 2 e. { 0 , 1 , 2 } |
24 |
23
|
a1i |
|- ( ph -> 2 e. { 0 , 1 , 2 } ) |
25 |
13 18 21 24
|
fnimatp |
|- ( ph -> ( <" I J K "> " { 0 , 1 , 2 } ) = { ( <" I J K "> ` 0 ) , ( <" I J K "> ` 1 ) , ( <" I J K "> ` 2 ) } ) |
26 |
|
s3fv0 |
|- ( I e. D -> ( <" I J K "> ` 0 ) = I ) |
27 |
1 26
|
syl |
|- ( ph -> ( <" I J K "> ` 0 ) = I ) |
28 |
|
s3fv1 |
|- ( J e. D -> ( <" I J K "> ` 1 ) = J ) |
29 |
2 28
|
syl |
|- ( ph -> ( <" I J K "> ` 1 ) = J ) |
30 |
|
s3fv2 |
|- ( K e. D -> ( <" I J K "> ` 2 ) = K ) |
31 |
3 30
|
syl |
|- ( ph -> ( <" I J K "> ` 2 ) = K ) |
32 |
27 29 31
|
tpeq123d |
|- ( ph -> { ( <" I J K "> ` 0 ) , ( <" I J K "> ` 1 ) , ( <" I J K "> ` 2 ) } = { I , J , K } ) |
33 |
15 25 32
|
3eqtrd |
|- ( ph -> ( <" I J K "> " dom <" I J K "> ) = { I , J , K } ) |
34 |
4 33
|
eqtr3id |
|- ( ph -> ran <" I J K "> = { I , J , K } ) |