Let $L = \{\langle M \rangle \mid \text{M is a TM which accepts only the string "010"}\}$. Prove that $L$ is undecidable.
This is my solution, reducing $A_{TM}$ to $L$:
$R(\langle M,w \rangle)$ outputs $M'(x)$, such that:
if $(x == "010")$: run $M$ on $w$ and do on $x$ what $M$ does on $w$
else reject $x$
The language $L(M')=\{010\} \iff M$ accepts $w$. Is the mapping reduction well done? Thank you.