A student made an error while proving the following claim that he thought was true.
Claim: Let $f : X \rightarrow Y$ be a function, and let $A$ and $B$ be subsets of $X$. Then $f ( A \cap B ) = f ( A ) \cap f ( B )$.
The student's proof: If I show that the sets $f ( A \cap B )$ and $f ( A ) \cap f ( B )$ are subsets of each other, the proof is complete.
Now let $c \in f ( A \cap B )$.\\
I. There exists a $d \in A \cap B$ such that $c = f ( d )$.\\
II. Since $d \in A$ and $d \in B$, we have $f ( d ) \in f ( A )$ and $f ( d ) \in f ( B )$. Thus $c = f ( d ) \in f ( A ) \cap f ( B )$.
On the other hand, let $c \in f ( A ) \cap f ( B )$.\\
III. We have $c \in f ( A )$ and $c \in f ( B )$. From this, there exists an $a \in A$ such that $c = f ( a )$ and a $\mathrm { b } \in \mathrm { B }$ such that $c = f ( b )$.\\
IV. Since $c = f ( a )$ and $c = f ( b )$, we have $a = b$.\\
V. Since $a \in A , b \in B$ and $a = b$, we have $a \in A \cap B$ and thus $c = f ( a ) \in f ( A \cap B )$.
In which of the numbered steps did this student make an error?\\
A) I\\
B) II\\
C) III\\
D) IV\\
E) V