Conditional expectation of Bernoulli random variable

Question

how can I prove that for $X$ and $Y$ random variables we have that $$ \mathbb{E}[X \vert Y] = P(X = 1 \vert Y) $$ if $X$ is a Bernoulli random variable?

Answer 1

$E(X|Y)=(0)P(X=0|Y)+(1)P(X=1|Y)=P(X=1|Y)$