In my "Foundations to Pure Mathematics" module, one of the slides, talking about Bijective functions defines it as follows:
Let $X$ and $Y$ be sets, and let $f: X \to Y$ be a function. Then $f$ is bijective if $f$ is both injective and surjective. This means that, for each $y \in Y$, there is exactly one point $x \in X$ with $f(x) = y$. A bijection is a bijective function.
He then follows this definition by writing: $\forall y \in Y \exists ! x \in X: f(x) = y$
Does the "!" in this context have any significant meaning, or is it just emphasizing the $\exists$?
It means there exists a unique $x$. It is shorthand for $$\exists x\left(( \operatorname{stuff}(x)) \wedge (\forall y (\operatorname{stuff}(y)) \implies x=y)\right)$$