Having trouble proving a language is NP-complete
Here is a more technical answer.  The "trick" is to put all the
work in the reduction function.

Assume $NP = P$, $L \in P$ and let $x_0, x_1$ be two instances, such
that $x_1 \in L$ and $x_0 \not\in L$.  We have to show that some
$NP$-complete problem $Q$ is poly-time reducible to $L$.  By
assumption, there is an algorithm $M$ deciding $Q$ in polynomial
time. 

Define a function
$$
f(x) =
\begin{cases}
  x_1 &amp; \mbox{if $M(x) = 1$} \\
  x_0 &amp; \mbox{else}
\end{cases}\mbox{ ,}
$$
for all instances $x$ of $Q$.  Then clearly $f \in FP$ and $x \in Q
\Leftrightarrow f(x) \in L$.