Proof of substitution rule for conditional expectation

Question

Let $v: \mathbb{R}^2 \to \mathbb{R}$ be a function and $X, Y$ random variables. It holds $$ \mathbb{E}[v(X,Y)|Y=y]=\mathbb{E}[v(X,y)|Y=y], \ y\in R(Y). $$

What would be a way to start the proof?

I know that I should post what I've tried but my attempts have failed miserably, so I would like a hint...

Answer 1

Given $Y(\omega)=y$, the random variables $V(X(\omega),Y(\omega))=V(X(\omega),y)=U(X(\omega))$, with probability measure $\mathbb{P}_{\left. X\,\right\vert Y=y}(\mathrm{d}\omega)$, thus the substitution rule.