My question is based off a book by $\mathbf{Erhan Cinlar - Probability\; and\; Stochastics, Chapter\;VI - theorem\; 3.2}$
$\mathbf{Some\;definitions:}$
$\mathbf{First}$, for an arbitrary index set $T$, the sub-$\sigma$-algebras $\mathcal{H_t},t\in T$ are said to be conditionally independent given $\mathcal{H}$ if $\mathbb{E}(V_{t_1}...V_{t_n}|\mathcal{H})=\mathbb{E}(V_{t_1}|\mathcal{H})...\mathbb{E}(V_{t_n}|\mathcal{H})$ for all integers $n\geq2$ and choices $t_1,...,t_n$ in $T$, $V_{t}\in\mathcal{H_t}$ is positive random variable for each $t\in T$. Random variables $Z_t,t\in T$, are said to be conditionally independent given $\mathcal{H}$ if the $\sigma$-algebras they generate are so given $\mathcal{H}$. If $\mathcal{H}=\sigma(W)$ for some collection of random variables $\{W_i,i\in L\}$ where $L$ can be an arbitrary index, then "given $\mathcal{H}$" is replaced by "given $W$".
$\mathbf{Second}$, let $(E,\mathcal{E})$ and $(F,\mathcal{F})$ be measurable spaces. Let $K$ be a mapping from $E$x$\mathcal{F}$ into $\mathbb{\bar{R}_+}$. Then $K$ is called a transition Kernel from $(E,\mathcal{E})$ into $(F,\mathcal{F})$ if:
a) the mapping $x\rightarrow K(x,B)$ is $\mathcal{E}$-measurable for every set $B\in\mathcal{F}$, and
b) the mapping $B\rightarrow K(x,B)$ is a measure on $(F,\mathcal{F})$ for every $X$ in $E$
If $K(x,F)=1$ for every $x$ in $E$, the kernel is called a transition probability kernel.
$\mathbf{Setup:}$
Let $(E,\mathcal{E})$ and $(F,\mathcal{F})$ be measurable spaces. Let $X=\{X_i:i\in I\}$ and $Y=\{Y_i:i \in I\}$ be collections, indexed by the same coutable set $I$, of random variables taking values in $(E,\mathcal{E})$ and $(F,\mathcal{F})$ respectively. The domain of both $X$ and $Y$ are from measurable space $(\Omega, \mathcal{G},\mathbb{P})$, with $\mathbb{P}$ being a probability measure.
$\mathbf{Question:}$ Suppose we have a transition probability Kernel from $(E,\mathcal{E})$ into $(F,\mathcal{F})$. Assume that given $X$, the variables $Y_i$ are conditionally independent and have the respective distributions $Q(X_i,\cdot)$. Consider a positive real-valued function $f$ in $\mathcal{E}\bigotimes\mathcal{F}$
Then \begin{align} \mathbb{E}\{\Pi_{i\in I} e^{-f(X_i,Y_i)} |\;X \}=\Pi_{i\in I}\int_FQ(X_i,dy)e^{-f(X_i,y)} \end{align}
I do not know why this equality holds. Can someone kindly explain it step by step?