Theorem imaundi

Description: Distributive law for image over union. Theorem 35 of Suppes p. 65. (Contributed by NM, 30-Sep-2002)

	Ref	Expression
Assertion	imaundi	$$⊢A\left[B\cup C\right]=A\left[B\right]\cup A\left[C\right]$$

Proof

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