# 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 $f(\varnothing) = \varnothing$?

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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 $\varnothing=f(\varnothing)$.

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