Confusion with contraposition of an implication with quantifiers I'm learning about functions and more specific, surjections and injections. However I'm a little bit confused regarding the use of quantifiers and "contraposition". 
Example: 
A function $ f: X \rightarrow Y$ is injective if  $ \forall x_1, x_2 \in X: x_1 \not = x_2 \implies f(x_1) \not = f(x_2) $, My book says this is equivalent to the contrapostion, that is: $ \forall x_1, x_2 \in X: f(x_1)  = f(x_2) \implies x_1 = x_2 $. So in this case the quantifiers don't change. 
However if we look at this example, which is an excercise my professor wrote out for us. 
The original implication is: $$\forall B \in P(Y): \forall C \in P(Y): f^{-1}(B) \subset f^{-1}(C) \implies B \subset C \implies f $$ is surjective
And our professor said that the contraposition of this implication is: $f$ is not surjective $$ \implies  \exists B \in P(Y): \exists C \in P(Y): \neg \left[ f^{-1}(B) \subset f^{-1}(C) \implies B \subset C \right] $$
So in this case the quantifiers do change. I can't seem to find the reason why sometimes they change and other times they don't. If anyone could clarify, I would highly appreciate it! 

Let $P$ be the statement  $\forall B \forall C [f^{-1}(B) \subset f^{-1}(C) \rightarrow B \subset C]$.
Let $Q$ be the statement "$f$ is surjective."
Hence, by substitution $P \rightarrow Q$ is logically equivalent to
$\forall B \forall C [f^{-1}(B) \subset f^{-1}(C) \rightarrow B \subset C] \Rightarrow (f$ is surjective$)$ 
The law of contraposition states
$$P \rightarrow Q \Leftrightarrow \neg Q \rightarrow \neg P$$
Hence, by substitution we have 
$\forall B \forall C [f^{-1}(B) \subset f^{-1}(C) \rightarrow B \subset C] \Rightarrow (f$ is surjective$)$ 
$\Leftrightarrow$
$\neg (f$ is surjective$)$ $ \Rightarrow \neg \forall B \forall C [f^{-1}(B) \subset f^{-1}(C) \rightarrow B \subset C] $ 
and by Demorgan's Law for Quantifers we have
$\neg (f$ is surjective$)$ $ \Rightarrow  \exists B \exists C \neg [f^{-1}(B) \subset f^{-1}(C) \rightarrow B \subset C] $
which is the same as
$(f$ is not surjective$)$ $ \Rightarrow  \exists B \exists C \neg [f^{-1}(B) \subset f^{-1}(C) \rightarrow B \subset C] $
