Problem from Pinter’s book

[Cratylus]

Let E and F be partially ordered classes, and let
g : E → F be an isomorphism. Prove that for
[MATH] x \in E[/MATH] g(S_x)=S_g(x)
conclude that S_x [MATH]\cong[/MATH] S_g(x).

questions
The question is assuming I showed each
function is an isomorphism, can I infer
equality ? If not, how about comparing
dom(g) and ran(g) of each function,

I have proved it by set inclusion but am not
happy with I t.