[proofplan] The proof is a direct continuous-mapping argument. We view the empirical process indexed by $\mathcal F$ as a random element $G_n\in\ell^\infty(\mathcal F)$, use the $P$-Donsker hypothesis to obtain $G_n\xrightarrow{d}G_P$, and then apply the continuous mapping principle to the deterministic pullback map $\Phi$. Finally, we identify $\Phi(G_n)$ pointwise with the reindexed empirical process and identify $\Phi(G_P)$ with $G_P\circ\psi$. [/proofplan]

[step:Define the empirical process as an element of $\ell^\infty(\mathcal F)$] For each $n\in\mathbb N$, define the empirical process \begin{align*} G_n:\mathcal F&\to\mathbb R \end{align*} by \begin{align*} G_n(f):=\sqrt n\{P_nf-Pf\}. \end{align*} The assumption that $\mathcal F$ is $P$-Donsker means precisely that, under the adopted empirical-process measurability convention, \begin{align*} G_n\xrightarrow{d}G_P \end{align*} in $\ell^\infty(\mathcal F)$. [/step]

[step:Apply the continuous mapping principle to the pullback map]By hypothesis, $\Phi:\ell^\infty(\mathcal F)\to\ell^\infty(T)$ is continuous for the supremum norms. Applying the continuous mapping principle, in the form recorded in [citetheorem:9834], to the convergence $G_n\xrightarrow{d}G_P$ and to the deterministic continuous map $\Phi$, we obtain \begin{align*} \Phi(G_n)\xrightarrow{d}\Phi(G_P) \end{align*} in $\ell^\infty(T)$.[/step]

[guided]We now transfer convergence from the original index set $\mathcal F$ to the new index set $T$. For each $n\in\mathbb N$, the empirical process is the map \begin{align*} G_n:\mathcal F&\to\mathbb R \end{align*} defined by \begin{align*} G_n(f):=\sqrt n\{P_nf-Pf\} \end{align*} for $f\in\mathcal F$. The input convergence is \begin{align*} G_n\xrightarrow{d}G_P \end{align*} in $\ell^\infty(\mathcal F)$, which is exactly the convergence supplied by the $P$-Donsker assumption. The map through which we pass this convergence is \begin{align*} \Phi:\ell^\infty(\mathcal F)&\to\ell^\infty(T), \end{align*} where, for each $z\in\ell^\infty(\mathcal F)$ and $t\in T$, \begin{align*} \Phi(z)(t):=z(\psi(t)). \end{align*} The continuous mapping principle requires convergence in distribution in the domain space and continuity of the deterministic map at the relevant limit points. The first requirement is the $P$-Donsker convergence. The second requirement is exactly the stated continuity hypothesis on $\Phi$ with respect to the supremum norms. Therefore, by the continuous mapping principle, stated for empirical-process statistics in [citetheorem:9834], the transformed random elements converge: \begin{align*} \Phi(G_n)\xrightarrow{d}\Phi(G_P) \end{align*} in $\ell^\infty(T)$. If the $P$-Donsker convergence is formulated using outer probability rather than Borel random elements, the same outer-probability version of the continuous mapping principle is being used here, so no additional measurability assertion is introduced at this step.[/guided]

[step:Identify the transformed empirical process pointwise] For each $n\in\mathbb N$ and each $t\in T$, the definition of $\Phi$ gives \begin{align*} \Phi(G_n)(t)=G_n(\psi(t)). \end{align*} Since $\psi(t)\in\mathcal F$, the definition of $G_n$ gives \begin{align*} G_n(\psi(t))=\sqrt n\{P_n(\psi(t))-P(\psi(t))\}. \end{align*} Thus $\Phi(G_n)$ is exactly the $T$-indexed process \begin{align*} t\mapsto \sqrt n\{P_n(\psi(t))-P(\psi(t))\}. \end{align*} Similarly, for each $t\in T$, \begin{align*} \Phi(G_P)(t)=G_P(\psi(t)), \end{align*} so $\Phi(G_P)=G_P\circ\psi$ as an element of $\ell^\infty(T)$. [/step]

[step:Conclude the reindexed convergence] Substituting the pointwise identifications from the previous step into \begin{align*} \Phi(G_n)\xrightarrow{d}\Phi(G_P) \end{align*} yields \begin{align*} \left(t\mapsto \sqrt n\{P_n(\psi(t))-P(\psi(t))\}\right)\xrightarrow{d}\left(t\mapsto G_P(\psi(t))\right) \end{align*} in $\ell^\infty(T)$. This is the asserted continuous mapping principle for the reindexed empirical process. [/step]