The use of the term "such that" confuses me I've seen this like $A=\{(x,y) :x,y\in\Bbb R\ \text{and } P(x,y) \}$ and $B=\{(x,y)\in \Bbb R^2:P(x,y)\}$ for some predicate $P$.
Is there any difference between those uses of $\text{such that}$? In other words, is $B=A$ true?
Another example: $X=\{x\in C\}, X'=\{x: x\in C\}$.
They are the same. $\{X \; : \; P(X)\}$ is the set of all $X$ that satisfy the condition $P(X)$. You may consider $$\{X \in A \; : \; P(X)\}$$ as a convenient abbreviation for $$\{X \; : \; X \in A \ \text{and}\ P(X) \}$$ Similarly, $$\{f(X) \; : \; P(X)\}$$ is a convenient abbreviation for $$\{Y: Y = f(X)\ \text{for some}\ X\ \text{such that}\ P(X)\}$$