analysis - Quick Clarification: Definition of Bijective Function

I am very familiar with the concepts of bijective, surjective and injective maps but I am interested in improvising the definition of bijection in a way I have not seen done before. To be clear I will provide the definitions that I use for surjective, injective and bijective although they are the common and generally accepted definitions.

$f:A \to B$ is $surjective$ $\iff$ $\forall y \in B$, $\exists x \in A$ : $f(x) = y$.

$f:A \to B$ is $injective$ $\iff$ $\forall x, y \in A,$ $f(x)=f(y) \implies x=y$

$f:A \to B$ is $bijective$ $\iff$ ($\forall y \in B$, $\exists x \in A$ : $f(x) = y$) $\wedge$ ($\forall x, y \in A,$ $f(x)=f(y) \implies x=y$)

You can see that the definition provided for bijection is simply the conjunction of surjective and injection.

My question is, does it suffice to define bijection as such:

$f:A \to B$ is $bijective$ $\iff$ $\forall y \in B$, $\exists! x \in A$ : $f(x) = y$.

Sorry for making a mountain out of a mole hill with this question, the person I usually work with is not here tonight so I have to run my spit ball ideas by you guys! Thanks in advance.