Collection of all ordered pairs with fixed first coordinate is a proper class

Question

Let $a$ be a fixed set. Set $A = \{(a,B) : B = B\}$. How do I show that the class $A$ is proper? I think it should be since it is basically the same as the class of all sets. I was considering using the axiom of replacement with a function which sends $(x, y) \to y$ but how do I write this function out as a definite condition? And where do I send everything that isn't an ordered pair?

Answer 1

You can write down a formula $\varphi(x,y)$ saying "$x$ is a pair of the form $(a,y)$". This $\varphi$ serves as the function you want to apply replacement to.

Note that the property $\varphi$ you define needs only be functional on the set $A$, so you don't have to send non-pairs anywhere.