How to show that $F(s) = E\left ( 1 \left \{ y \leq s \right \} \right )$?

Question

For any $s$. Here 1 is an indicator function.

Also let $L(y) = F$, where $F$ is a CDF.

My attempt: $$E\left ( 1 \left \{ y \leq s \right \} \right ) = 1 \times P\left ( y \leq s \right ) + 0 \times P\left ( y > s \right ) = P\left ( y\leq s \right ) $$

Then how should I be proceeding??