Step |
Hyp |
Ref |
Expression |
1 |
|
s2rn.i |
|- ( ph -> I e. D ) |
2 |
|
s2rn.j |
|- ( ph -> J e. D ) |
3 |
|
imadmrn |
|- ( <" I J "> " dom <" I J "> ) = ran <" I J "> |
4 |
1 2
|
s2cld |
|- ( ph -> <" I J "> e. Word D ) |
5 |
|
wrdfn |
|- ( <" I J "> e. Word D -> <" I J "> Fn ( 0 ..^ ( # ` <" I J "> ) ) ) |
6 |
|
s2len |
|- ( # ` <" I J "> ) = 2 |
7 |
6
|
oveq2i |
|- ( 0 ..^ ( # ` <" I J "> ) ) = ( 0 ..^ 2 ) |
8 |
|
fzo0to2pr |
|- ( 0 ..^ 2 ) = { 0 , 1 } |
9 |
7 8
|
eqtri |
|- ( 0 ..^ ( # ` <" I J "> ) ) = { 0 , 1 } |
10 |
9
|
fneq2i |
|- ( <" I J "> Fn ( 0 ..^ ( # ` <" I J "> ) ) <-> <" I J "> Fn { 0 , 1 } ) |
11 |
10
|
biimpi |
|- ( <" I J "> Fn ( 0 ..^ ( # ` <" I J "> ) ) -> <" I J "> Fn { 0 , 1 } ) |
12 |
4 5 11
|
3syl |
|- ( ph -> <" I J "> Fn { 0 , 1 } ) |
13 |
12
|
fndmd |
|- ( ph -> dom <" I J "> = { 0 , 1 } ) |
14 |
13
|
imaeq2d |
|- ( ph -> ( <" I J "> " dom <" I J "> ) = ( <" I J "> " { 0 , 1 } ) ) |
15 |
|
c0ex |
|- 0 e. _V |
16 |
15
|
prid1 |
|- 0 e. { 0 , 1 } |
17 |
16
|
a1i |
|- ( ph -> 0 e. { 0 , 1 } ) |
18 |
|
1ex |
|- 1 e. _V |
19 |
18
|
prid2 |
|- 1 e. { 0 , 1 } |
20 |
19
|
a1i |
|- ( ph -> 1 e. { 0 , 1 } ) |
21 |
|
fnimapr |
|- ( ( <" I J "> Fn { 0 , 1 } /\ 0 e. { 0 , 1 } /\ 1 e. { 0 , 1 } ) -> ( <" I J "> " { 0 , 1 } ) = { ( <" I J "> ` 0 ) , ( <" I J "> ` 1 ) } ) |
22 |
12 17 20 21
|
syl3anc |
|- ( ph -> ( <" I J "> " { 0 , 1 } ) = { ( <" I J "> ` 0 ) , ( <" I J "> ` 1 ) } ) |
23 |
|
s2fv0 |
|- ( I e. D -> ( <" I J "> ` 0 ) = I ) |
24 |
1 23
|
syl |
|- ( ph -> ( <" I J "> ` 0 ) = I ) |
25 |
|
s2fv1 |
|- ( J e. D -> ( <" I J "> ` 1 ) = J ) |
26 |
2 25
|
syl |
|- ( ph -> ( <" I J "> ` 1 ) = J ) |
27 |
24 26
|
preq12d |
|- ( ph -> { ( <" I J "> ` 0 ) , ( <" I J "> ` 1 ) } = { I , J } ) |
28 |
14 22 27
|
3eqtrd |
|- ( ph -> ( <" I J "> " dom <" I J "> ) = { I , J } ) |
29 |
3 28
|
eqtr3id |
|- ( ph -> ran <" I J "> = { I , J } ) |