It is well known that the evolutionary dynamics characterizes the process of competition and evolution of phenotypes and behaviors in a population. Intuitively, the individual with a higher fitness will have a higher survival probability, which should be reflected in the evolutionary dynamic model. However, due to the computational complexity of fixation probability, it is very difficult to prove the existence of this property in evolutionary dynamics on complex networks. This paper aims at providing a rigorously theoretical proof for the global existence of such property in the local evolutionary dynamics by using the coupling and splicing techniques. In particular, we also prove that the fixation probability is monotone increasing for the initial nodes set of mutants. Numerical simulations are also given to validate the proposed approaches.