Let $f:X \to Y$ be measurable map between measurable spaces w.r.t. to their corresponding $\sigma$-algebras $\Sigma_X$ and $\Sigma_Y$, resp.

If $(X,\Sigma_X)$ is a standard Borel space can we always conclude that the graph: $$ \Gamma(f) := \{ (x,y) \in X \times Y \mid y=f(x) \} $$ is measurable, i.e. an element of the product $\sigma$-algebra $\Sigma_X \otimes \Sigma_Y$?

Note that I'm interested in the case where $(Y,\Sigma_Y)$ is a general measurable space and not restricted to e.g. being countably separated, which would imply the claim (even for general $(X,\Sigma_X)$).