The restriction of a variable in a Sobolev function is a.e. Sobolev Let $I,J$ be bounded open intervals in $\mathbb{R}$.

How to show that $W^{1,1}(I\times J) $ is embedded in the Bochner space $L^1(I, W^{1,1}(J))$?

Is there a standard reference where this is covered?
There is a natural map on $W^{1,1}(I\times J)$, which takes $x$ to $f_x=f(x,\cdot)$ which is a function on $J$.
I tried showing that the weak derivative $\frac{\partial f_x}{\partial  y}(y)$ exist and is equal to the given weak derivative $\frac{\partial f}{\partial  y}(x,y)$, but failed. I understand this is related to the Fubini-Tonelli theorem somehow, but I am not sure about the details.
Showing $\frac{\partial f_x}{\partial  y}(y)=\frac{\partial f}{\partial  y}(x,y)$ amounts to establishing the following equality,
$$ \int_J f(x,y) \phi'(y)dy=\int_J f_x \phi'(y)dy=-\int_J \frac{\partial f}{\partial  y}(x,y) \phi(y)dy$$
holds for a.e. $x \in I$. 
I would like to see a detailed argument or a reference.
 A: What do you think about that? Probably I am overlooking something.
Let $t \in I, x \in J$. If $f \in W^{1,1}(I \times J)$ then $f,\partial_t f,\partial_x f \in L^1(I \times J)$. Therefore it follows
$$\begin{aligned} \|f\|_{L^1(I,W^{1,1}(J))}=\int_I \|f_x(t)\|_{W^{1,1}(J)} dt &= \int_I \big( \|f_x(t)\|_{L^1(J)}  + \|\partial_x f_x(t)\|_{L^1(J)} \big)dt \\ &=\int_I \int_J f(t,x)+\partial_x f(t,x) dtdx \\ &= \|f\|_{L^1(I \times J)} + \|\partial_x f \|_{L^1(I \times J)}. \end{aligned}$$

EDIT: Okay, I get it. Let $[f_x(t)](x)=f(t,x)$. We know that $\partial_x f \in L^1(I \times J)$ which is defined by
$$ \int_{I \times J} \partial_x f(t,x) \phi(t,x) \ dtdx = - \int_{I \times J} f(t,x) \partial_x \phi(t,x) \ dtdx$$
for all $\phi \in C_c^\infty(I \times J).$ Or in other words
$$ \int_{I \times J} \partial_x [f_x(t)](x) \phi(t,x) \ dtdx = - \int_{I \times J} [f_x(t)](x) \partial_x \phi(t,x) \ dtdx$$
for all $\phi \in C_c^\infty(I \times J)$. Now we use that $C_c^\infty(I) \otimes C_c^\infty(J)$ is dense in $C_c^\infty(I \times J)$. Therefore we rewrite this as
$$ \int_{I} \left[ \int_J \partial_x [f_x(t)](x) \varphi(x) \ dx \right] \eta(t) dt = - \int_I \left[ \int_J [f_x(t)](x) \partial_x \varphi(x) \ dx \right] \eta(t) dt$$
for all $\varphi \in C_c^\infty(J)$, $\eta \in C_c^\infty(I)$. By the fundamental lemma of calculus of variations
$$ \int_J \partial_x [f_x(t)](x) \varphi(x) \ dx  = -  \int_J [f_x(t)](x) \partial_x \varphi(x) \ dx$$
for all $\varphi \in C_c^\infty(J)$ and almost every $t \in I$. Therefore $f_x(t) \in W^{1,1}(J)$ for a.e. $t \in I$ with the weak time derivative $\partial_x f_x(t) \in L^1(J)$.
