What is a good explanation for "$y = f(x)$ has the same meaning as $(x, y) ∈ f$"? From Pinter's, A Book of Set Theory, the statement "$y = f(x)$ has the same meaning as $(x, y) ∈ f$" is emboldened below Corollary 2.4.
The justification given for the same meaning between the two statements is that it is customary. But I don't know how to convince myself that in all applications, that these statements are equivalent.
The best justification I have is that we can be sure for a function $f: A\rightarrow B$, for every $x\in A$, we are given a unique element $y\in B$, so this symbolic leap is justified in the sense that it is unique and comparable to its notational change; that is we will always have a unique pair of statements for each $x\in A$.
But my justification feels like I'm handwaving. I have some doubts because we are dealing in different predicates, namely '$\in$' and '$=$', and just because I can't think of a situation where these alternative statements may not work does not alleviate my doubt over the potential for error.
Right now I'm operating on some faith that it just always works, but if someone were to be so kind to provide a proper explanation or some texts that might point me in the right direction, I would appreciate that.
 A: Strictly speaking, we define $f(x)$ to be the unique $y$ such that $(x,y) \in f \subset A \times B$, but that answer is pretty unilluminating, so I'll try to motivate the definition for you.
Sometimes, we say that a function $f:A \to B$ takes each element in its domain $A$ and maps it to an element in its range $f(A) \subset B$. However, there really isn't any need to think of a function as a dynamic process; functions are just a way of associating each element in some set $A$ with exactly one element of $B$. So, we can formalize the concept of a function as a subset of $A \times B$, where, for each $x \in A$, there is exactly one element $(x,y) \in A \times B$, namely, $(x,f(x))$. This definition carries all the information we need.
A: A function is defined in C. C. Pinters book as

*

*2.1 Definition: A function from $A$ to $B$ is a triple of objects $(f,A,B)$, where $A$ and $B$ are classes and $f$ is a subclass of $A\times B$ with the following properties.

*

*(F1) $\ \ \quad\forall x\in A,\exists y\in B$ such that $(x,y)\in f$.


*(F2) $\quad$  If $(x, y_1)\in f$ and $(x,y_2)\in f$, then $y_1=y_2$.
It is customary to write $f:A\to B$ instead of $(f,A,B)$.
Note the last statement indicates just a convenient syntactical replacement. When we look at the statement presented below Corollary 2.4 we can read:

*

*Let $f:A\to B$ be a function and let $x\in A$; it is customary to use the symbol $f(x)$ to designate the image of $x$. Thus, $y=f(x)$ has the same meaning as $(x,y)\in f$.

Again, this is nothing else than just a convenient syntactical replacement. It is immediately afterwards used by the author to present F1 and F2 in a more convenient notation. He states

*

*When we write $y=f(x)$ instead of $(x,y)\in f$, Conditions F1 and F2 take the form

*

*(F1) $\quad\ $ $\forall x\in A,\exists y\in B, y=f(x)$.


*(F2) $\quad$ If $y_1=f(x)$ and $y_2=f(x)$, then $y_1=y_2$.
This is precisely the same as F1 and F2 in definition 2.1 above just using a more convenient, the common notation.
A: You are reading Pinter's book. Let me quote from this book:
Beginning of quotation
We begin by giving our “official” definition of a function.
2.1 Definition A function from $A$ to $B$ is a triple of objects $\langle f, A, B \rangle$, where $A$ and $B$ are classes and $f$ is a subclass of $A × B$ with the following properties.
F1. $∀x ∈ A, ∃y ∈ B$ such that $(x, y) ∈ f$.
F2. If $(x, y_1) ∈ f$ and $(x, y_2) ∈ f$, then $y_1 = y_2$.
It is customary to write $f : A → B$ instead of $\langle f, A, B \rangle$.
In ordinary mathematical applications, every function $f : A → B$ is a function from a set $A$ to a set $B$. However, the intuitive concept of a function from $A$ to $B$ is meaningful for any two collections $A$ and $B$, whether $A$ and $B$ be sets or proper classes; hence it is natural to give the definition of a function in its most general form, letting $A$ and $B$ be any classes. Once again, every set is a class, hence everything we have to say about functions from a class $A$ to a class $B$ applies, in particular, to functions from a set $A$ to a set $B$.
Let $f : A → B$ be a function; if $(x, y) ∈ f$, we say that $y$ is the image of $x$ (with respect to $f$).
[...]
Let $f : A → B$ be a function and let $x ∈ A$; it is customary to use the
symbol $f(x)$ to designate the image of $x$. Thus,
$y = f(x)$ has the same meeting as $(x, y) ∈ f$.
End of quotation
Certainly "has the same meeting" is a typo and should be replaced by "has the same meaning" as you write in your question. The above compilation of the basic definition of a function and of the subsequent notational conventions shows that the following are equivalent:

*

*$(x,y) \in f$

*$y$ is the image of $x$ with respect to $f$ [notational convention]

*$f(x)  = y$ [notational convention]

