Proof verification : If $S$ and $T$ are subsets of $B$ with $S \subset T$, prove $f^{-1}(S)\subset f^{-1}(T)$

Question

EDT. Clarification: $f:A\rightarrow B$

EDT. Changed the first line in the proof according to responses / comments.

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Proof.

Let $a \in f^{-1}(S)$. Then, $f(a)\in S$.

Because $S\subset T$, if $f(a)\in S$ then also $f(a)\in T$ and therefore $a \in f^{-1}(T)$.

For all $a \in f^{-1}(S)$ we then have $a \in f^{-1}(T)$ and therefore $f^{-1}(S)\subset f^{-1}(T)$

QED

Answer 1

Your start is again wrong. You have to take $a\in f^{-1}(S)$ then you can say $f(a)\in S$ (and not vice versa).

Every time you want to prove $M\subseteq N$, you have to take $x\in M$ (and just in $M$, no $f(M)$ or $f^{-1}(M)$ or something else, just $M$)! Then you prove that $x$ is also in $N$.