Step |
Hyp |
Ref |
Expression |
1 |
|
reelprrecn |
|- RR e. { RR , CC } |
2 |
1
|
a1i |
|- ( ( F : CC --> CC /\ RR C_ dom ( CC _D F ) ) -> RR e. { RR , CC } ) |
3 |
|
simpl |
|- ( ( F : CC --> CC /\ RR C_ dom ( CC _D F ) ) -> F : CC --> CC ) |
4 |
|
ssidd |
|- ( ( F : CC --> CC /\ RR C_ dom ( CC _D F ) ) -> CC C_ CC ) |
5 |
|
simpr |
|- ( ( F : CC --> CC /\ RR C_ dom ( CC _D F ) ) -> RR C_ dom ( CC _D F ) ) |
6 |
|
dvres3 |
|- ( ( ( RR e. { RR , CC } /\ F : CC --> CC ) /\ ( CC C_ CC /\ RR C_ dom ( CC _D F ) ) ) -> ( RR _D ( F |` RR ) ) = ( ( CC _D F ) |` RR ) ) |
7 |
2 3 4 5 6
|
syl22anc |
|- ( ( F : CC --> CC /\ RR C_ dom ( CC _D F ) ) -> ( RR _D ( F |` RR ) ) = ( ( CC _D F ) |` RR ) ) |