In Terence Tao's book "Analysis 1", in definition 3.3.1 (function definition), he says
Let $X, Y$ be sets, and let $P(x, y)$ be a property pertaining to an object $x \in X$ and an object $y \in Y$, such that for every $x \in X$, there is exactly one $y \in Y$ for which $P(x, y)$ is true. Then we define the function $f: X \rightarrow Y$ defined by $P$ on the domain $X$ and range $Y$ to be the object which, given any input $x \in X$, assigns an output $f(x) \in Y$, defined to be the unique object $f(x)$ for which $P(x, f(x))$ is true.
This definition is on a chapter on set theory foundations, which starts by postulating the existence of objects and sets (he does impure set theory, so not all objects are sets).
Here, under the conditions of the definition, he says that there exists an object $f$, which has the property that for any $x\in X$, $f(x)$ denotes the unique $y\in Y$ such that $P(x,y)$ is true. Shouldn't this be an axiom that postulates the existence of an object?
In another question Equality of functions: axiom or definition? it is said that "Tao does mention that strictly speaking, this definition is an axiom that posits the existence of a function given sets $X,Y$ and property $P$." However, I did not find such a statement in the book.