Let $F :A \rightarrow B$ be a function and $A_i$ with $i \in I$ subsets of $A$.
First show that: $$ F(\cap_{i\in I} A_i) \subset \cap_{i \in I} F(A_i) $$
Then find an example where $\subsetneq$ applies.
Here's my proof:
Let $b \in F(\cap_{i \in I}A_i)$. Therefore $\exists a \in \cap_{i \in I}A_i$ such that $F(a) = b$. Since $a \in \cap_{i \in I}A_i$, it follows that $a \in A_i$ is valid $\forall i \in I$. Since: $$ F(A_i) = \{f(a) | a \in A_i\} $$ we can conclude that $b \in F(A_i) \forall i \in I$, and finally $b \in \cap_{i \in I}F(A_i)$.
I think I was able to prove it but I can't find an example with proper subset... Can someone please give me a hint or show me an example? Also, is my proof correct? If so, what would you change to improve it a.k.a make it more "elegant"?
Thank you!
The proof looks great, I wouldn't change it.
For counter example, take $f:\{0,1,2\}→\{0,1\}$ with $f(0)=0,f(1)=1,f(2)=1$.
Then $A_0=\{0,1\},A_1=\{0,2\},A_0\cap A_1=\{0\},f(\{0\})=\{0\},f(A_0)=\{0,1\},f(A_1)=\{0,1\},$$f(A_0)\cap f(A_1)=\{0,1\}$ and $\{0\}\subsetneq\{0,1\}$