Apparently, $\varphi (x,y) : y = P(x)$ where $P$ denotes the power set is not absolute for transitive models.
We call a formula $\varphi(v_1, \dots v_n)$ in $L_S$ absolute for a class $\mathbf X$ iff $\forall x_1, \dots , x_n \in \mathbf X (\varphi (x_1 , \dots , x_n) \leftrightarrow \varphi_{/\mathbf X}(x_1, \dots , x_n ))$ where $\varphi_{/\mathbf X}$ denotes the formula $\varphi$ with all occurrences of $\exists v_i$ replaces by $\exists v_i \in \mathbf X$ and similarly for $\forall$.
In words it means that the formula is true regardless of whether we quantify over the whole universe $\mathbf V$ or only over elements of $\mathbf X$.
It is easy to break $\varphi (x,y) : y = P(x)$ in non-transitive models: for example if $M = \{ a, \{\varnothing , a\}, \{\varnothing , a, b\} \}$ then in $M$, $\{\varnothing , a, b\}$ is a power set of $a$ but outside $M$ this is false.
Can someone show me a transitive model in which $\varphi (x,y) : y = P(x)$ is not absolute? Many thanks for your help.