The image of the union is the union of the images

up:: Image of Function

Let $f:X\to Y$ be a function and subsets $A,B\subset X$.

Then let $y\in f(A\cup B)$ a point in $f$‘s Image. Then

$$y\in f(A\cup B)⟺\exists x\in A\cup B\mid f(x)=y⟺(\exists x\in A\mid f(x)=y)\vee (\exists x\in B\mid f(x)=y)$$

By the definition of the image of a function, we have that

$$y\in f(A\cup B)⟺y\in f(A)\vee y\in f(B)⟺y\in f(A)\cup f(B)$$

Thus, $f(A\cup B)\subseteq f(A)\cup f(B)$ and $f(A\cup B)\supseteq f(A)\cup f(B)$.

Therefore, $f(A\cup B)=f(A)\cup f(B)$.