Theorem imaundir

Description: The image of a union. (Contributed by Jeff Hoffman, 17-Feb-2008)

	Ref	Expression
Assertion	imaundir	$$⊢\left(A\cup B\right)\left[C\right]=A\left[C\right]\cup B\left[C\right]$$

Proof

Step	Hyp	Ref	Expression
1		df-ima	$$⊢\left(A\cup B\right)\left[C\right]=\mathrm{ran}\left({\left(A\cup B\right)\upharpoonright }_C\right)$$
2		resundir	$$⊢{\left(A\cup B\right)\upharpoonright }_C=\left({A\upharpoonright }_C\right)\cup \left({B\upharpoonright }_C\right)$$
3	2	rneqi	$$⊢\mathrm{ran}\left({\left(A\cup B\right)\upharpoonright }_C\right)=\mathrm{ran}\left(\left({A\upharpoonright }_C\right)\cup \left({B\upharpoonright }_C\right)\right)$$
4		rnun	$$⊢\mathrm{ran}\left(\left({A\upharpoonright }_C\right)\cup \left({B\upharpoonright }_C\right)\right)=\mathrm{ran}\left({A\upharpoonright }_C\right)\cup \mathrm{ran}\left({B\upharpoonright }_C\right)$$
5	1 3 4	3eqtri	$$⊢\left(A\cup B\right)\left[C\right]=\mathrm{ran}\left({A\upharpoonright }_C\right)\cup \mathrm{ran}\left({B\upharpoonright }_C\right)$$
6		df-ima	$$⊢A\left[C\right]=\mathrm{ran}\left({A\upharpoonright }_C\right)$$
7		df-ima	$$⊢B\left[C\right]=\mathrm{ran}\left({B\upharpoonright }_C\right)$$
8	6 7	uneq12i	$$⊢A\left[C\right]\cup B\left[C\right]=\mathrm{ran}\left({A\upharpoonright }_C\right)\cup \mathrm{ran}\left({B\upharpoonright }_C\right)$$
9	5 8	eqtr4i	$$⊢\left(A\cup B\right)\left[C\right]=A\left[C\right]\cup B\left[C\right]$$