| Step |
Hyp |
Ref |
Expression |
| 1 |
|
mhmcopsr.p |
|- P = ( I mPwSer R ) |
| 2 |
|
mhmcopsr.q |
|- Q = ( I mPwSer S ) |
| 3 |
|
mhmcopsr.b |
|- B = ( Base ` P ) |
| 4 |
|
mhmcopsr.c |
|- C = ( Base ` Q ) |
| 5 |
|
mhmcopsr.h |
|- ( ph -> H e. ( R MndHom S ) ) |
| 6 |
|
mhmcopsr.f |
|- ( ph -> F e. B ) |
| 7 |
|
fvexd |
|- ( ph -> ( Base ` S ) e. _V ) |
| 8 |
|
ovex |
|- ( NN0 ^m I ) e. _V |
| 9 |
8
|
rabex |
|- { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin } e. _V |
| 10 |
9
|
a1i |
|- ( ph -> { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin } e. _V ) |
| 11 |
|
eqid |
|- ( Base ` R ) = ( Base ` R ) |
| 12 |
|
eqid |
|- ( Base ` S ) = ( Base ` S ) |
| 13 |
11 12
|
mhmf |
|- ( H e. ( R MndHom S ) -> H : ( Base ` R ) --> ( Base ` S ) ) |
| 14 |
5 13
|
syl |
|- ( ph -> H : ( Base ` R ) --> ( Base ` S ) ) |
| 15 |
|
eqid |
|- { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin } = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin } |
| 16 |
1 11 15 3 6
|
psrelbas |
|- ( ph -> F : { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin } --> ( Base ` R ) ) |
| 17 |
14 16
|
fcod |
|- ( ph -> ( H o. F ) : { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin } --> ( Base ` S ) ) |
| 18 |
7 10 17
|
elmapdd |
|- ( ph -> ( H o. F ) e. ( ( Base ` S ) ^m { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin } ) ) |
| 19 |
|
reldmpsr |
|- Rel dom mPwSer |
| 20 |
19 1 3
|
elbasov |
|- ( F e. B -> ( I e. _V /\ R e. _V ) ) |
| 21 |
6 20
|
syl |
|- ( ph -> ( I e. _V /\ R e. _V ) ) |
| 22 |
21
|
simpld |
|- ( ph -> I e. _V ) |
| 23 |
2 12 15 4 22
|
psrbas |
|- ( ph -> C = ( ( Base ` S ) ^m { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin } ) ) |
| 24 |
18 23
|
eleqtrrd |
|- ( ph -> ( H o. F ) e. C ) |