What is an image of empty set? Let $f: M \rightarrow N$ be a function. We always can define image of custom set $A \subset M$ like $f(A) = \{y\in{N}:(f(x) = y) \wedge(x\in A) \}$. As known, empty set is a subset of each set, so my question is: Is it proper to write $f(\varnothing) = \varnothing$?
Well, $y\in f(A)$ if and only if $y=f(a)$ for some $a\in A$.

However there are no elements in the empty set, so there is no $y\in f(\varnothing)$, therefore $\varnothing=f(\varnothing)$.
The notation {x∈X:P(x)} denotes the set of all elements x belonging to X for which the proposition P is true.  

The truth table for "∧" shows (f(x)=y)∧(x∈A) is true only when both (f(x)=y) and (x∈A) are both T.

But if A=∅, then (x∈A) is always F.

As such, the proposition P in your definition [i.e. (f(x)=y)∧(x∈A) ] is always F ... and so the set of all elements y belonging to N for which the proposition P is true is empty.