I have a problem with a proof: I need to prove that $\Gamma=\lbrace(x,y)| f(x)=y\rbrace $ is measurable, where $f:X\rightarrow Y$ is a measurable function from a metric space $X$ with its Borel $\sigma$-algebra into $(Y,\mathcal{F})$ that is a measurable space with $\mathcal{F}$ countable generated and that contains singleton.
I'm reading a proof where they first find a countable algebra of sets $\mathscr{U}$ that generates $\mathcal{F}$, then they show that for every $y\in Y$ $$ \lbrace y\rbrace=\bigcap_{B\in\mathscr{U},y\in B}\ B. $$ By this they conclude that $$\Gamma=\bigcap_{B\in\mathscr{U}}\left\lbrace (x,y)\mid x\in f^{-1}(B),y\in B \right\rbrace $$ is measurable, but the right-hand side it's empty and this equality seems to me not true.
There is a good way to write the graph that ensure me measurability?