In this article, we have proposed a wider class of distributions by modifying the distribution function of the baseline density. This new class is a generalization of many well-known generators such as beta family, Kumaraswamy family, Kummer beta generalized family and Topp-Leone family. Furthermore, we have introduced a subcase, known as G-Fixed-Topp-Leone class, with different properties and have provided the expression for the reliability in the multicomponent stress-strength model. Additionally, we have studied the exponential-fixed-Topp-Leone distribution as an example; some structural properties of this three-parameter exponential distribution are driven which also include the derivations of incomplete moments, mean deviation, measures of uncertainty, reliability in multicomponent stress-stress model, order statistics, Lorenz, Bonferroni and Zenga curves. The estimation of the unknown parameters is done by the method of maximum likelihood. We have also included a real-life application of this new three-parameter exponential distribution to two datasets. A numerical study for the reliability in the multicomponent stress-strength model for the exponential-fixed-Topp-Leone distribution, using the Markov Chain and Monte Carlo (MCMC) method, is also performed.