Set $Z = V(f_1, \ldots, f_r)$. It is clear that $Z \to X$ is locally of finite presentation, hence the composition $Z \to Y$ is locally of finite presentation, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-composition-finite-presentation}. Hence it suffices to show that $Z \to Y$ is flat in a neighbourhood of $p(z)$. Let $k \subset k'$ be an extension field. Then $F' = F \times_{\Spec(k)} \Spec(k')$ is surjective and flat over $F$, hence we can find a point $z' \in |F'|$ mapping to $z$ and the local ring map $\mathcal{O}_{F, \overline{z}} \to \mathcal{O}_{F', \overline{z}'}$ is flat, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-flat-at-point-etale-local-rings}. Hence the image of $f_1, \ldots, f_r$ in $\mathcal{O}_{F', \overline{z}'}$ is a regular sequence too, see Algebra, Lemma \ref{algebra-lemma-flat-increases-depth}. Thus, during the proof we may replace $k$ by an extension field. In particular, we may assume that $z \in |F|$ comes from a section $z : \Spec(k) \to F$ of the structure morphism $F \to \Spec(k)$. \medskip\noindent Choose a scheme $V$ and a surjective \'etale morphism $V \to Y$. Choose a scheme $U$ and a surjective \'etale morphism $U \to X \times_Y V$. After possibly enlarging $k$ once more we may assume that $\Spec(k) \to F \to X$ factors through $U$ (as $U \to X$ is surjective). Let $u : \Spec(k) \to U$ be such a factorization and denote $v \in V$ the image of $u$. Note that the morphisms $$ U_v \times_{\Spec(\kappa(v))} \Spec(k) = U \times_V \Spec(k) \to U \times_Y \Spec(k) \to F $$ are \'etale (the first as the base change of $V \to V \times_Y V$ and the second as the base change of $U \to X$). Moreover, by construction the point $u : \Spec(k) \to U$ gives a point of the left most space which maps to $z$ on the right. Hence the elements $f_1, \ldots, f_r$ map to a regular sequence in the local ring on the right of the following map $$ \mathcal{O}_{U_v, u} \longrightarrow \mathcal{O}_{U_v \times_{\Spec(\kappa(v)} \Spec(k), \overline{u}} = \mathcal{O}_{U \times_V \Spec(k), \overline{u}}. $$ But since the displayed arrow is flat (combine More on Flatness, Lemma \ref{flat-lemma-flat-up-down-henselization} and Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-flat-at-point-etale-local-rings}) we see from Algebra, Lemma \ref{algebra-lemma-flat-increases-depth} that $f_1, \ldots, f_r$ maps to a regular sequence in $\mathcal{O}_{U_v, u}$. By More on Morphisms, Lemma \ref{more-morphisms-lemma-slice-given-elements} we conclude that the morphism of schemes $$ V(f_1, \ldots, f_r) \times_X U = V(f_1|_U, \ldots, f_r|_U) \to V $$ is flat in an open neighbourhood $U'$ of $u$. Let $X' \subset X$ be the open subspace corresponding to the image of $|U'| \to |X|$ (see Properties of Spaces, Lemmas \ref{spaces-properties-lemma-topology-points} and \ref{spaces-properties-lemma-open-subspaces}). We conclude that $V(f_1, \ldots, f_r) \cap X' \to Y$ is flat (see Morphisms of Spaces, Definition \ref{spaces-morphisms-definition-flat}) as we have the commutative diagram $$ \xymatrix{ V(f_1, \ldots, f_r) \times_X U' \ar[d]_a \ar[r] & V \ar[d]^b \\ V(f_1, \ldots, f_r) \cap X' \ar[r] & Y } $$ with $a, b$ \'etale and $a$ surjective.