Theorem dmun

Description: The domain of a union is the union of domains. Exercise 56(a) of Enderton p. 65. (Contributed by NM, 12-Aug-1994) (Proof shortened by Andrew Salmon, 27-Aug-2011)

	Ref	Expression
Assertion	dmun	$$⊢\mathrm{dom}\left(A\cup B\right)=\mathrm{dom}A\cup \mathrm{dom}B$$

Proof

Step	Hyp	Ref	Expression
1		unab	$$⊢\left\{y|\exists xyAx\right\}\cup \left\{y|\exists xyBx\right\}=\left\{y|\left(\exists xyAx\vee \exists xyBx\right)\right\}$$
2		brun	$$⊢y\left(A\cup B\right)x\leftrightarrow \left(yAx\vee yBx\right)$$
3	2	exbii	$$⊢\exists xy\left(A\cup B\right)x\leftrightarrow \exists x\left(yAx\vee yBx\right)$$
4		19.43	$$⊢\exists x\left(yAx\vee yBx\right)\leftrightarrow \left(\exists xyAx\vee \exists xyBx\right)$$
5	3 4	bitr2i	$$⊢\left(\exists xyAx\vee \exists xyBx\right)\leftrightarrow \exists xy\left(A\cup B\right)x$$
6	5	abbii	$$⊢\left\{y|\left(\exists xyAx\vee \exists xyBx\right)\right\}=\left\{y|\exists xy\left(A\cup B\right)x\right\}$$
7	1 6	eqtri	$$⊢\left\{y|\exists xyAx\right\}\cup \left\{y|\exists xyBx\right\}=\left\{y|\exists xy\left(A\cup B\right)x\right\}$$
8		df-dm	$$⊢\mathrm{dom}A=\left\{y|\exists xyAx\right\}$$
9		df-dm	$$⊢\mathrm{dom}B=\left\{y|\exists xyBx\right\}$$
10	8 9	uneq12i	$$⊢\mathrm{dom}A\cup \mathrm{dom}B=\left\{y|\exists xyAx\right\}\cup \left\{y|\exists xyBx\right\}$$
11		df-dm	$$⊢\mathrm{dom}\left(A\cup B\right)=\left\{y|\exists xy\left(A\cup B\right)x\right\}$$
12	7 10 11	3eqtr4ri	$$⊢\mathrm{dom}\left(A\cup B\right)=\mathrm{dom}A\cup \mathrm{dom}B$$