This is a follow-up question to this question
I proved the following identity. $\Omega \subset \mathbb{R}^d$ is open and $g$ is a metric field on $\Omega$. Further $\Gamma_{\,kl}^j$ denotes the Christoffel symbol of second kind. $g^{ij}$ is the inverse matrix to $g_{ij}$ and $g$ denotes the determinant.
$$ \Gamma_{\,jl}^j = \frac{\partial}{\partial x^l} \log \sqrt{\vert g \vert} $$
Now my question is:
Is it trivial to state that this equation is invariant under diffeomorphisms i.e.
$$ \Gamma_{\,jl}^j[\phi] = \frac{\partial}{\partial x^l} \log \sqrt{\vert g[\phi] \vert} $$
where $\phi \in$ Diff$(\Omega)$? If not, how would I go about proving that?
The Christoffel Symbols transform in a rather difficult manner, however because of two indices being the same $(j)$ two of the $g_{ij}$ terms cancel each other out when we write down the definition of the Christoffel symbol, which reduces the problem to $$ \frac{1}{2}g[\phi]^{jk}\frac{\partial}{\partial x^l}g[\phi]_{jk} = \frac{\partial}{\partial x^l} \log \sqrt{\vert g[\phi] \vert} $$