Theorem bj-pr2un

Description: The second projection preserves unions. (Contributed by BJ, 6-Apr-2019)

	Ref	Expression
Assertion	bj-pr2un	$$⊢pr2\left(A\cup B\right)=pr2A\cup pr2B$$

Proof

Step	Hyp	Ref	Expression
1		bj-projun	$$⊢\left(1_{𝑜}Proj\left(A\cup B\right)\right)=\left(1_{𝑜}ProjA\right)\cup \left(1_{𝑜}ProjB\right)$$
2		df-bj-pr2	$$⊢pr2\left(A\cup B\right)=\left(1_{𝑜}Proj\left(A\cup B\right)\right)$$
3		df-bj-pr2	$$⊢pr2A=\left(1_{𝑜}ProjA\right)$$
4		df-bj-pr2	$$⊢pr2B=\left(1_{𝑜}ProjB\right)$$
5	3 4	uneq12i	$$⊢pr2A\cup pr2B=\left(1_{𝑜}ProjA\right)\cup \left(1_{𝑜}ProjB\right)$$
6	1 2 5	3eqtr4i	$$⊢pr2\left(A\cup B\right)=pr2A\cup pr2B$$