Step |
Hyp |
Ref |
Expression |
1 |
|
mstaval.r |
|- R = ( mStRed ` T ) |
2 |
|
mstaval.s |
|- S = ( mStat ` T ) |
3 |
|
eqid |
|- ( mPreSt ` T ) = ( mPreSt ` T ) |
4 |
3 1
|
msrf |
|- R : ( mPreSt ` T ) --> ( mPreSt ` T ) |
5 |
|
ffn |
|- ( R : ( mPreSt ` T ) --> ( mPreSt ` T ) -> R Fn ( mPreSt ` T ) ) |
6 |
|
fvelrnb |
|- ( R Fn ( mPreSt ` T ) -> ( X e. ran R <-> E. s e. ( mPreSt ` T ) ( R ` s ) = X ) ) |
7 |
4 5 6
|
mp2b |
|- ( X e. ran R <-> E. s e. ( mPreSt ` T ) ( R ` s ) = X ) |
8 |
3
|
mpst123 |
|- ( s e. ( mPreSt ` T ) -> s = <. ( 1st ` ( 1st ` s ) ) , ( 2nd ` ( 1st ` s ) ) , ( 2nd ` s ) >. ) |
9 |
8
|
fveq2d |
|- ( s e. ( mPreSt ` T ) -> ( R ` s ) = ( R ` <. ( 1st ` ( 1st ` s ) ) , ( 2nd ` ( 1st ` s ) ) , ( 2nd ` s ) >. ) ) |
10 |
|
id |
|- ( s e. ( mPreSt ` T ) -> s e. ( mPreSt ` T ) ) |
11 |
8 10
|
eqeltrrd |
|- ( s e. ( mPreSt ` T ) -> <. ( 1st ` ( 1st ` s ) ) , ( 2nd ` ( 1st ` s ) ) , ( 2nd ` s ) >. e. ( mPreSt ` T ) ) |
12 |
|
eqid |
|- ( mVars ` T ) = ( mVars ` T ) |
13 |
|
eqid |
|- U. ( ( mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) = U. ( ( mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) |
14 |
12 3 1 13
|
msrval |
|- ( <. ( 1st ` ( 1st ` s ) ) , ( 2nd ` ( 1st ` s ) ) , ( 2nd ` s ) >. e. ( mPreSt ` T ) -> ( R ` <. ( 1st ` ( 1st ` s ) ) , ( 2nd ` ( 1st ` s ) ) , ( 2nd ` s ) >. ) = <. ( ( 1st ` ( 1st ` s ) ) i^i ( U. ( ( mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( ( mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) ) , ( 2nd ` ( 1st ` s ) ) , ( 2nd ` s ) >. ) |
15 |
11 14
|
syl |
|- ( s e. ( mPreSt ` T ) -> ( R ` <. ( 1st ` ( 1st ` s ) ) , ( 2nd ` ( 1st ` s ) ) , ( 2nd ` s ) >. ) = <. ( ( 1st ` ( 1st ` s ) ) i^i ( U. ( ( mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( ( mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) ) , ( 2nd ` ( 1st ` s ) ) , ( 2nd ` s ) >. ) |
16 |
9 15
|
eqtrd |
|- ( s e. ( mPreSt ` T ) -> ( R ` s ) = <. ( ( 1st ` ( 1st ` s ) ) i^i ( U. ( ( mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( ( mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) ) , ( 2nd ` ( 1st ` s ) ) , ( 2nd ` s ) >. ) |
17 |
4
|
ffvelrni |
|- ( s e. ( mPreSt ` T ) -> ( R ` s ) e. ( mPreSt ` T ) ) |
18 |
16 17
|
eqeltrrd |
|- ( s e. ( mPreSt ` T ) -> <. ( ( 1st ` ( 1st ` s ) ) i^i ( U. ( ( mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( ( mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) ) , ( 2nd ` ( 1st ` s ) ) , ( 2nd ` s ) >. e. ( mPreSt ` T ) ) |
19 |
12 3 1 13
|
msrval |
|- ( <. ( ( 1st ` ( 1st ` s ) ) i^i ( U. ( ( mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( ( mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) ) , ( 2nd ` ( 1st ` s ) ) , ( 2nd ` s ) >. e. ( mPreSt ` T ) -> ( R ` <. ( ( 1st ` ( 1st ` s ) ) i^i ( U. ( ( mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( ( mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) ) , ( 2nd ` ( 1st ` s ) ) , ( 2nd ` s ) >. ) = <. ( ( ( 1st ` ( 1st ` s ) ) i^i ( U. ( ( mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( ( mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) ) i^i ( U. ( ( mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( ( mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) ) , ( 2nd ` ( 1st ` s ) ) , ( 2nd ` s ) >. ) |
20 |
18 19
|
syl |
|- ( s e. ( mPreSt ` T ) -> ( R ` <. ( ( 1st ` ( 1st ` s ) ) i^i ( U. ( ( mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( ( mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) ) , ( 2nd ` ( 1st ` s ) ) , ( 2nd ` s ) >. ) = <. ( ( ( 1st ` ( 1st ` s ) ) i^i ( U. ( ( mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( ( mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) ) i^i ( U. ( ( mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( ( mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) ) , ( 2nd ` ( 1st ` s ) ) , ( 2nd ` s ) >. ) |
21 |
|
inass |
|- ( ( ( 1st ` ( 1st ` s ) ) i^i ( U. ( ( mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( ( mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) ) i^i ( U. ( ( mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( ( mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) ) = ( ( 1st ` ( 1st ` s ) ) i^i ( ( U. ( ( mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( ( mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) i^i ( U. ( ( mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( ( mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) ) ) |
22 |
|
inidm |
|- ( ( U. ( ( mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( ( mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) i^i ( U. ( ( mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( ( mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) ) = ( U. ( ( mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( ( mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) |
23 |
22
|
ineq2i |
|- ( ( 1st ` ( 1st ` s ) ) i^i ( ( U. ( ( mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( ( mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) i^i ( U. ( ( mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( ( mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) ) ) = ( ( 1st ` ( 1st ` s ) ) i^i ( U. ( ( mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( ( mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) ) |
24 |
21 23
|
eqtri |
|- ( ( ( 1st ` ( 1st ` s ) ) i^i ( U. ( ( mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( ( mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) ) i^i ( U. ( ( mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( ( mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) ) = ( ( 1st ` ( 1st ` s ) ) i^i ( U. ( ( mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( ( mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) ) |
25 |
24
|
a1i |
|- ( s e. ( mPreSt ` T ) -> ( ( ( 1st ` ( 1st ` s ) ) i^i ( U. ( ( mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( ( mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) ) i^i ( U. ( ( mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( ( mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) ) = ( ( 1st ` ( 1st ` s ) ) i^i ( U. ( ( mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( ( mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) ) ) |
26 |
25
|
oteq1d |
|- ( s e. ( mPreSt ` T ) -> <. ( ( ( 1st ` ( 1st ` s ) ) i^i ( U. ( ( mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( ( mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) ) i^i ( U. ( ( mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( ( mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) ) , ( 2nd ` ( 1st ` s ) ) , ( 2nd ` s ) >. = <. ( ( 1st ` ( 1st ` s ) ) i^i ( U. ( ( mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( ( mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) ) , ( 2nd ` ( 1st ` s ) ) , ( 2nd ` s ) >. ) |
27 |
20 26
|
eqtrd |
|- ( s e. ( mPreSt ` T ) -> ( R ` <. ( ( 1st ` ( 1st ` s ) ) i^i ( U. ( ( mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( ( mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) ) , ( 2nd ` ( 1st ` s ) ) , ( 2nd ` s ) >. ) = <. ( ( 1st ` ( 1st ` s ) ) i^i ( U. ( ( mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( ( mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) ) , ( 2nd ` ( 1st ` s ) ) , ( 2nd ` s ) >. ) |
28 |
16
|
fveq2d |
|- ( s e. ( mPreSt ` T ) -> ( R ` ( R ` s ) ) = ( R ` <. ( ( 1st ` ( 1st ` s ) ) i^i ( U. ( ( mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) X. U. ( ( mVars ` T ) " ( ( 2nd ` ( 1st ` s ) ) u. { ( 2nd ` s ) } ) ) ) ) , ( 2nd ` ( 1st ` s ) ) , ( 2nd ` s ) >. ) ) |
29 |
27 28 16
|
3eqtr4d |
|- ( s e. ( mPreSt ` T ) -> ( R ` ( R ` s ) ) = ( R ` s ) ) |
30 |
|
fveq2 |
|- ( ( R ` s ) = X -> ( R ` ( R ` s ) ) = ( R ` X ) ) |
31 |
|
id |
|- ( ( R ` s ) = X -> ( R ` s ) = X ) |
32 |
30 31
|
eqeq12d |
|- ( ( R ` s ) = X -> ( ( R ` ( R ` s ) ) = ( R ` s ) <-> ( R ` X ) = X ) ) |
33 |
29 32
|
syl5ibcom |
|- ( s e. ( mPreSt ` T ) -> ( ( R ` s ) = X -> ( R ` X ) = X ) ) |
34 |
33
|
rexlimiv |
|- ( E. s e. ( mPreSt ` T ) ( R ` s ) = X -> ( R ` X ) = X ) |
35 |
7 34
|
sylbi |
|- ( X e. ran R -> ( R ` X ) = X ) |
36 |
1 2
|
mstaval |
|- S = ran R |
37 |
35 36
|
eleq2s |
|- ( X e. S -> ( R ` X ) = X ) |