Let $f: X → Y$, $x ↦ f(x)$ be a mapping and $X$ and $Y$ sets. As mentioned in the title, my question is, if therefore the following statement is true.
$$∀f(x) ∈ Y ∃! x ∈ X ⇔ ∀a,b ∈ X : f(a) = f(b) ⇒ a = b$$
That is, is $f$ injective, if and only if $∀f(x) ∈ Y ∃! x ∈ X$ is true?