Is image of recursive set under recursive function recursive?

Question

Given $A$ - recursive set and function $f$ which is also recursive. Is $f(A)$ recursive?

I think that it isn't recursive, but how to prove it?

Answer 1

Do the usual sort of indexing of arithmetical sentences, and let our set $S$ be the (recursive) set of indices of proofs, say in first-order Peano arithmetic.

For any proof with index $n$, let $f(n)$ be the index of the last sentence in the proof (the theorem). Then $f(S)$ is not recursive.