Step |
Hyp |
Ref |
Expression |
1 |
|
rnghmf1o.b |
|- B = ( Base ` R ) |
2 |
|
rnghmf1o.c |
|- C = ( Base ` S ) |
3 |
|
isrngisom |
|- ( ( R e. V /\ S e. W ) -> ( F e. ( R RngIsom S ) <-> ( F e. ( R RngHomo S ) /\ `' F e. ( S RngHomo R ) ) ) ) |
4 |
1 2
|
rnghmf1o |
|- ( F e. ( R RngHomo S ) -> ( F : B -1-1-onto-> C <-> `' F e. ( S RngHomo R ) ) ) |
5 |
4
|
bicomd |
|- ( F e. ( R RngHomo S ) -> ( `' F e. ( S RngHomo R ) <-> F : B -1-1-onto-> C ) ) |
6 |
5
|
a1i |
|- ( ( R e. V /\ S e. W ) -> ( F e. ( R RngHomo S ) -> ( `' F e. ( S RngHomo R ) <-> F : B -1-1-onto-> C ) ) ) |
7 |
6
|
pm5.32d |
|- ( ( R e. V /\ S e. W ) -> ( ( F e. ( R RngHomo S ) /\ `' F e. ( S RngHomo R ) ) <-> ( F e. ( R RngHomo S ) /\ F : B -1-1-onto-> C ) ) ) |
8 |
3 7
|
bitrd |
|- ( ( R e. V /\ S e. W ) -> ( F e. ( R RngIsom S ) <-> ( F e. ( R RngHomo S ) /\ F : B -1-1-onto-> C ) ) ) |