A PL embedded polyhedron becomes a subcomplex after subdivision

By [F1], choose a finite linear triangulation $T_0$ of $Q$ in which $f(P)$ is a subcomplex.

Regard $T$ and $T_0$ as finite linear complexes on the same embedded polyhedron $Q$. Form all nonempty cells $\sigma \cap \tau$ for $\sigma \in T$ and $\tau \in T_0$, together with their faces. By [F2] these form a finite convex cell complex covering $Q$, and a compatible finite simplicial triangulation $T^{\mathrm{\prime }}$ of that cell complex exists. Each simplex of $T^{\mathrm{\prime }}$ lies in a cell contained in a simplex of $T$ and a simplex of $T_0$, so $T^{\mathrm{\prime }}$ is a common linear refinement. In particular it is a subdivision of $T$. Every simplex of the subcomplex representing $f(P)$ is a union of these refined simplices; thus the simplices of $T^{\mathrm{\prime }}$ contained in $f(P)$ have realization exactly $f(P)$ and form a subcomplex by [F3].