Simple predicate question

Question

I know if a set $A$ is $A=\{x\mid P(x)\}$, then $x\in A$ if and only if $P(x)$. My qeustion is what is $A=\{x\in B\mid P(x)\}$? $x\in B$ and $P(x)$ if anf only if $x\in A$?

Answer 1

$$A = \{ x \in B: P(x) \}$$ is shorthand for $$A = \{x: \text{$x \in B$ and $P(x)$} \}$$

That is, $x \in A$ if and only if $x \in B$ and $P(x)$.