Image e and preimage of a set using logical connectors

Question

This might appear silly, I know that

$$ \begin{array}{l} f(A) = \left\{f(x) : x \in A \right\} \\ f^{-1}(A) = \left\{x : f(x) \in A \right\} \end{array} $$

If $y\in f(A)$ is fixed can this be expressed by the condition

$$ \exists x, x\in A \Rightarrow f(x)=y $$

Can we do a similar thing with the pre-image?

Basically I'd like to express those definition using logical connectors. Is this possible?

Answer 1

$y\in f(A)$ is equivalent to $$ \exists x\,(x\in A\land f(x)=y)$$ and $x\in f^{-1}(A)$ is equivalent to $$ f(x)\in A.$$

Answer 2

$y\in f(A)$ is equivalent with: $\exists x\;[x\in A\wedge f(x)=y]$.

So $\wedge$ instead of $\implies$.

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We have the (very useful) equivalence:$$x\in f^{-1}(A)\iff f(x)\in A$$

So if $x\in f^{-1}(A)$ is fixed then this can be expressed by the condition $f(x)\in A$.

Does this meet your needs?