My understanding is that a function mapping a set $A$ to a set $B$ is defined as a relation $f \subseteq A \times B$ such that for every $x \in X$ there exists a unique $y \in Y$ such that $(x, y) \in f$.
Is it possible to use this definition to show that for any $x, x' \in X$, $x = x' \implies f(x) = f(x')$?
(1)x=x' ("x" and "x'"denote the same element of A) ;(2)f(x)= f(x) (an axiom for equality or obvious ) .When variables denote the same object one may substitute one variable for the other in any occurrence which is free for substitution ( not in the scope of a quantifier ) Thus substituting "x'" for "x" for the second occurrence in (2) gives f(x)=f(x') as desired .