Step |
Hyp |
Ref |
Expression |
1 |
|
eqid |
|- ( Base ` R ) = ( Base ` R ) |
2 |
|
eqid |
|- ( Base ` S ) = ( Base ` S ) |
3 |
1 2
|
rhmf |
|- ( F e. ( R RingHom S ) -> F : ( Base ` R ) --> ( Base ` S ) ) |
4 |
|
frel |
|- ( F : ( Base ` R ) --> ( Base ` S ) -> Rel F ) |
5 |
|
dfrel2 |
|- ( Rel F <-> `' `' F = F ) |
6 |
4 5
|
sylib |
|- ( F : ( Base ` R ) --> ( Base ` S ) -> `' `' F = F ) |
7 |
3 6
|
syl |
|- ( F e. ( R RingHom S ) -> `' `' F = F ) |
8 |
|
id |
|- ( F e. ( R RingHom S ) -> F e. ( R RingHom S ) ) |
9 |
7 8
|
eqeltrd |
|- ( F e. ( R RingHom S ) -> `' `' F e. ( R RingHom S ) ) |
10 |
9
|
anim1ci |
|- ( ( F e. ( R RingHom S ) /\ `' F e. ( S RingHom R ) ) -> ( `' F e. ( S RingHom R ) /\ `' `' F e. ( R RingHom S ) ) ) |
11 |
|
isrim0 |
|- ( F e. ( R RingIso S ) <-> ( F e. ( R RingHom S ) /\ `' F e. ( S RingHom R ) ) ) |
12 |
|
isrim0 |
|- ( `' F e. ( S RingIso R ) <-> ( `' F e. ( S RingHom R ) /\ `' `' F e. ( R RingHom S ) ) ) |
13 |
10 11 12
|
3imtr4i |
|- ( F e. ( R RingIso S ) -> `' F e. ( S RingIso R ) ) |