Summary: A subgraph of an edge-colored graph is called rainbow if all of its edges have different colors. For a graph \(G\) and a positive integer \(n\), the anti-Ramsey number \(ar(n,G)\) is the maximum number of colors in an edge-coloring of \(K_n\) with no rainbow copy of \(H\). Anti-Ramsey numbers were introduced by \textit{P. Erdős} et al. [in: Infinite and finite sets. To Paul Erdős on his 60th birthday. Vols. I, II, III. Amsterdam-London: North-Holland Publishing Company. 633--643 (1975; Zbl 0316.05111)] and studied in numerous papers. Let \(G\) be a graph with anti-Ramsey number \(\operatorname{ar}(n,G)\). In this paper we show the lower bound for \(\operatorname{ar}(n,pG)\), where \(pG\) denotes \(p\) vertex-disjoint copies of \(G\). Moreover, we prove that in some special cases this bound is sharp.