Let $V\subseteq\mathbb C^n$ be an irreducible affine variety, then the coordinate ring $$\mathbb C[V] = \mathbb C[x_1,\dots,x_n]\big/\mathbf I(V)$$ is an integral domain. Let $f\in\mathbb C[V]\setminus\{0\}$, then we can define the localization $$ \mathbb C[V]_f = \left\{\,g\big/f^\ell \in \mathbb C(V)\,\big|\, g\in\mathbb C[V], \ell\ge 0\,\right\}, $$ where $\mathbb C(V)$ denotes the field of fractions of $\mathbb C[V]$.
I want to proof that $\mathbb C[V]_f$ is the coordinate ring of the principal open subset $$V_f = \left\{\,p\in V\,\big|\, f(p)\neq 0\,\right\}.$$
We can see $V_f$ as an affine variety by identifying it with $$\widetilde{V_f} = \mathbf V(\mathbf I(V)+\langle gy-1\rangle) \subseteq \mathbb C^n\times \mathbb C,$$ where $y$ is $(n+1)$th coordinate, $g\in\mathbb C[x_1,\dots,x_n]$ represents $f\in\mathbb C[V]$ and the projection $\mathbb C^n\times \mathbb C\to\mathbb C^n$ maps $\widetilde{V_f}$ bijectively onto $V_f$. Then \begin{align} \mathbb C[V_f] &\cong \mathbb C[\widetilde{V_f}] = \mathbb C[x_1,\dots,x_n,y]\big/(\mathbf I(V)+\langle gy-1\rangle)\\ &\cong \mathbb C\left[x_1,\dots,x_n,1\big/g\right]\big/\mathbf I(V)\\ &\cong \left(\mathbb C[V]\right)\left[1\big/f\right] \cong \mathbb C[V]_f. \end{align} Is this reasoning correct? I'm not sure everything done in the last chain of isomorphisms is rigorous.
Yes it's correct.
For details see, for example, these notes definition 1.13 and the example right before definition 1.16.