Suppose I have two random variables $X$ and $Y$ which is not independent. Also, I have a function $g:\mathbb{R}^2\rightarrow \mathbb{R}$ and I know that for any fixed $y\in \mathbb{R}$ we have $$\mathbb{E}[g(X,y)]=h(y),$$ where $h$ Is known. Is it right to say
$$\mathbb{E}[g(X,Y)|Y=y]=h(y)?$$ This sounds correct to me but I don’t know how to justify it theoretically. Thank you very much in advance.
Counterexample: $Y=X$ with $X$ symmetric, e.g., $X\sim\cal N(0,1)$. Take $g(x,y):=xy$ and $h(y):\equiv0$. Then $$\Bbb E[g(X,y)]=y\,\Bbb E[X]=0=h(y)$$ for any $y\in\Bbb R$, but $$\Bbb E[g(X,Y)\mid Y=y]=y^2\neq h(y).$$