Discussion on Function Definitions and Related Axioms Within the Coq Proof Assistant

Axiomatic set theory forms the foundation of modern mathematics, and the Morse-Kelley axiomatic set theory system is more convenient for applications compared to the ZFC system and the NBG system. This text is based on the Morse-Kelley axiomatic set theory. It defines the intersection and union of members in a way that differs from the naive set theory. It then further explores the relevant definitions of functions as well as the axiom of substitution and the axiom of amalgamation, establishing a more comprehensive machine-assisted proof system. Function is one of the fundamental concept in set theory, and the formalization of function provides great convenience for the subsequent establishment of axiomatic systems for real numbers and exponent operations. The entire formalization process in this paper has been verified by Coq and run successfully on computer, with the proof process having standardization and reliability.