We present a study of the process $e^+e^-topi^{pm}(Dbar{D}^*)^{mp}$ using data samples of 1092~pb$^{-1}$ at $sqrt{s}=4.23$~GeV and 826~pb$^{-1}$ at $sqrt{s}=4.26$~GeV collected with the BESIII detector at the BEPCII storage ring. With full reconstruction of the $D$ meson pair and the bachelor $pi^{pm}$ in the final state, we confirm the existence of the charged structure $Z_c(3885)^{mp}$ in the $(Dbar{D}^*)^{mp}$ system in the two isospin processes $e^+e^-topi^+D^0D^{*-}$ and $e^+e^-topi^+D^-D^{*0}$. By performing a simultaneous fit, the statistical significance of $Zc(3885)^{mp}$ signal is determined to be greater than 10$sigma$, and its pole mass and width are measured to be $M_{rm{pole}}$=(3881.7$pm$1.6(stat.)$pm$1.6(syst.))~MeV/$c^2$ and $Gamma_{rm{pole}}$=(26.6$pm$2.0(stat.)$pm$2.1(syst.))~MeV, respectively. The Born cross section times the $(Dbar{D}^*)^{mp}$ branching fraction ($sigma(e^+e^-topi^{pm}Z_{c}(3885)^{mp}) times Br(Z_{c}(3885)^{mp}to(Dbar{D}^*)^{mp})$) is measured to be $(141.6pm7.9(text{stat.})pm12.3(text{syst.}))~text{pb}$ at $sqrt{s}=4.23$~GeV and $(108.4pm6.9(text{stat.})pm8.8(text{syst.}))~text{pb}$ at $sqrt{s}=4.26$~GeV. The polar angular distribution of the $pi^{pm}$-$Z_c(3885)^{mp}$ system is consistent with the expectation of a quantum number assignment of $J^P=1^+$ for $Z_c(3885)^{mp}$.
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