One-to-one function theorem in the hyper-reals
Note on notation: This seems pretty standard, but just to be clear, for any function F defined on the reals (R), *F is the extension of F to the hyper-reals (*R). This basically means that any logical sentence true of F in the reals is true of *F in the hyper-reals. If you're familiar with the hyper-reals, you should be aware of this relation.
I need to prove the following:
Let F: A -> R where A is some subset of R, and let F be one-to-one. We've already shown this implies *F: *A -> *R. Show that if we have an x such that x is in *A but x is not in A, then *F(x) is not in R (but of course it is in *R).
This makes intuitive sense to me but I have no idea how to show it.
Any help is appreciated, even if you can't totally solve it. Just showing me another avenue of thought will surely be helpful.
Thanks in advance.