Applying fundamental theorem of calculus where upper integration bound for F's associated integral is not equal to F's input

Normally, the FTC is stated like this:

Fundamental Theorem of Calculus: Let $f$ be Riemann integrable on $[a, b]$. For $x \in$$[a, b]$, define $F(x)=\int_a^x f$. If $f$ is continuous at a point $x \in(a, b)$, then $F$ is differentiable at $x$, and $F^{\prime}(x)=f(x)$.

However, I'm looking at the following solution to a homework problem:

Question 6.2.26. Let $U \subset \mathbb{R}^n$ be a connected open set, and let $\omega$ be a continuous differential 1-form on U. Suppose that $\int_\phi \omega=\int_\psi \omega$ whenever $\phi$ and $\psi$ begin and end at the same points. Show that there exists a $C^1$ function $f: U \rightarrow \mathbb{R}$ such that $\omega=d f$. (Hint: Consider how the fundamental theorem of calculus allows us to construct the antiderivative of a continuous function of a single variable as an integral.)

Let $p_0 \in U$ be fixed. Given any $p \in U$, we know that there exists some $\gamma_p:\left[a_p, b_p\right] \rightarrow \mathbb{R}^n$ such that $\gamma_p$ is a smooth parametrized curve and $\gamma_p\left(a_p\right)=p_0$ and $\gamma_p\left(b_p\right)=p$. We then define$$f(p):=\int_{\gamma_p} \omega=\int_{a_p}^{b_p} \omega_{\gamma_p(t)}\left(\gamma_p^{\prime}(t)\right) d t$$which is well-defined by our assumption that the line integral of $\omega$ is path independent.Our goal is now to show that $\omega=d f$. Given any point $x \in U$ and $v \in \mathbb{R}^n$, we want to show that $\omega_x(v)=d f_x(v)$. We define a path $\kappa:(-\varepsilon, \varepsilon) \rightarrow U$ given by$$\kappa(t)=x+t v$$

We have$$f(\kappa(t))=\int_{\gamma_{\kappa(t)}(s)} \omega=\int_{a_{\kappa(t)}}^{b_{\kappa(t)}} \omega_{\gamma_{\kappa(t)}(s)}\left(\gamma_{\kappa(t)}^{\prime}(s)\right) d s .$$

By the fundamental theorem of calculus, we have$$\frac{d}{d t}\left(f(\kappa(t))=\omega_{\kappa(t)}(v) .\right.$$

Then for $x=\kappa(0)$, we check that $d f_x(v)=\omega_x(v)$ as desired. Since $\omega$ is continuous and differentiable, we can check that $f$ is a $C^1$ function.

My question is this -- how is this an application of the Fundamental Theorem of Calculus? It seems like since f takes an input different from the upper integration bound, we cannot use the Fundamental Theorem; how is this resolved?