I have the following problem.
Let $p:Y\to X$ be a bundle with fiber $F$. This means that for each $x\in X$, there is an open neighborhood $N_x$ of $x$ and a homeomorphism $p^{-1}N_x\cong N_x\times F$ such that $$(p^{-1}N_x\xrightarrow{\cong} N_x\times F\xrightarrow{\pi_1} N_x)=(p^{-1}N_x\xrightarrow{p}N_x).$$ Let $f:A\to X$ be a continuous function. Let $\Gamma_f=\{(a,f(a)):a\in A\}$ and $\eta_{f,p}=\{(a,v):f(a)=p(v)\}$. Then $(id_A\times p)|^{\eta_{f,p}}_{\Gamma_{f}}:\eta_{f,p}\to\Gamma_f$ is a bundle with fiber $F$ which is what we want.
I said suppose $(a,f(a))\in\Gamma_f$. Then note that there exists an open neighborhood $N_{f(a)}$ of $f(a)$ and a homeomorphism $p^{-1}N_{f(a)}\cong N_{f(a)}\times F$ such that $$(p^{-1}N_{f(a)}\xrightarrow{\cong} N_{f(a)}\times F\xrightarrow{\pi_1} N_{f(a)})=(p^{-1}N_{f(a)}\xrightarrow{p}N_{f(a)}).$$ Since $f$ is a continuous function, $f^{-1}N_{f(a)}$ is open. I then decided to look at $(f^{-1}N_{f(a)}\times N_{f(a)})\cap \Gamma_f$ and said $((id_A\times p)|^{\eta_{f,p}}_{\Gamma_{f}})^{-1}((f^{-1}N_{f(a)}\times N_{f(a)})\cap \Gamma_f)=\eta_{f,p}\cap (f^{-1}N_{f(a)}\times p^{-1}N_{f(a)})$.I believe that $\eta_{f,p}\cong \Gamma_f\times F$. If this is true, then $\eta_{f,p}\cap (f^{-1}N_{f(a)}\times p^{-1}N_{f(a)})=(f^{-1}N_{f(a)}\times p^{-1}N_{f(a)})\cap \eta_{f,p}\cong f^{-1}N_{f(a)}\times(N_{f(a)}\times F)\cap (\Gamma_f \times F)\cong ((f^{-1}N_{f(a)}\times N_{f(a)})\times F\cup\Gamma_f\times F=((f^{-1}N_{f(a)}\times N_{f(a)})\cap\Gamma_f)\times F$.
I do not know how to proceed from here. Any hints would be helpful.
Note that I am working out the details of a problem in Vector Bundles and K-theory. The link for the book is here.
$\eta_{f, p} \to \Gamma_f$ is the pullback of the bundle $p: Y \to X$ along the map $\pi_2: \Gamma_f \to X$ which sends $(a, f(a)) \mapsto f(a)$. If $U \subseteq X$ is an open neighborhood such that $p|_{p^{-1}(U)}: p^{-1}(U) \to U$ is a trivial fiber bundle, then it is not hard to see $\eta_{f, p} \to \Gamma_f$ is trivial over $\pi_2^{-1}(U)$. Similarly, you can check that the transition maps on fibers are still elements of the same structure group.