Induced connection is well-defined

Question

Let $M=$Riemannian manifold,$N=$differentiable manifold, $\phi:N\to M$ be smooth map.

If $v\in T_xM$, and $\{E_i\}_{i=1}^n$ is a frame field in a neighborhood $V$ of $\phi(x)\in M$, then $$\forall x\in \phi^{-1}(V)\subset N, X(x)=X^i(x)E_i(\phi(x))\in T_{\phi(x)}M$$ for some functions $X^i(x)$ on $N$. We define $$\bar{D}_vX=v(X^i)(x)E_i(\phi(x))+X^i(x)D_{d\phi(v)}E_i$$ where $D$=Levi-Civita connection on $M$.

Then I try to show $\bar{D}_vX$ is well-defined, that is, it's independent of the choice of $\{E_i\}_{i=1}^n$.

This is the first time I deal with change of frame, following is what I have done.

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Let $\{F_i\}_{i=1}^n$, let $X(x)=Y^i(x)F_i(\phi(x))$ and $F_i=C^j_iE_j$, where $C^j_i$ are smooth functions on $M$. Then $Y^i=X^j(C^{-1})^i_j$, where $(C^{-1})^i_j$ is inverse matrix for $C^j_i$.

By definition, w.r.t $\{F_i\}_{i=1}^n$,

$$\begin{align} \bar{D}_vX&=v(Y^i)(x)F_i(\phi(x))+Y^i(x)D_{d\phi(v)}F_i\\ &=v(X^k(C^{-1})^i_k)(x)C^j_iE_j+X^k(C^{-1})^i_kD_{d\phi(v)}C^j_iE_j\\ &=[v(X^k)(C^{-1})^i_k+X^kv((C^{-1})^i_k)]C^j_iE_j+X^k(C^{-1})^i_k[(d\phi(v)C^j_i)E_i+C^j_iD_{d\phi(v)}E_j]\\ &=v(X^k)(C^{-1})^i_kC^j_iE_j+X^kv((C^{-1})^i_k)C^j_iE_j+X^k(C^{-1})^i_k(d\phi(v)C^j_i)E_j+X^k(C^{-1})^i_kC^j_iD_{d\phi(v)}E_j\\ &=v(X^k)\delta^j_kE_j+X^k[v((C^{-1})^i_k)C^j_i+(C^{-1})^i_k(d\phi(v)C^j_i)]E_j+X^k\delta^j_kD_{d\phi(v)}E_j\\ &=v(X^i)(x)E_i(\phi(x))+X^i(x)D_{d\phi(v)}E_i+X^k[v((C^{-1})^i_k)C^j_i+(C^{-1})^i_k(d\phi(v)C^j_i)]E_j \end{align}$$

Then consider $$v((C^{-1})^i_k)C^j_i+(C^{-1})^i_k(d\phi(v)C^j_i)$$ I hope to show $v((C^{-1})^i_k)C^j_i+(C^{-1})^i_k(d\phi(v)C^j_i)=v((C^{-1})^i_kC^j_i)=v(\delta^j_k)=0$, then I am done.

Question:

1. Is my proof (in particular the last step) correct?

2. It seems $C^j_i$ is smooth functions on $M$ (since it's transformation of frame on $M$), and $(C^{-1})^i_j$ is smooth functions on $N$ (since it transforms coefficiences $X^i$, which are functions on $N$). And $(C^{-1})^i_j$ is inverse of $C^j_i$, will this lead to a contradiction? What's the domain of $C^j_i$,$(C^{-1})^i_j$? And how do both $d\phi(v)C^j_i, v(C^i_k)$ make sense?