Theorem bj-pr2un

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

	Ref	Expression
Assertion	bj-pr2un	⊢ pr2 ( 𝐴 ∪ 𝐵 ) = ( pr2 𝐴 ∪ pr2 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		bj-projun	⊢ ( 1o Proj ( 𝐴 ∪ 𝐵 ) ) = ( ( 1o Proj 𝐴 ) ∪ ( 1o Proj 𝐵 ) )
2		df-bj-pr2	⊢ pr2 ( 𝐴 ∪ 𝐵 ) = ( 1o Proj ( 𝐴 ∪ 𝐵 ) )
3		df-bj-pr2	⊢ pr2 𝐴 = ( 1o Proj 𝐴 )
4		df-bj-pr2	⊢ pr2 𝐵 = ( 1o Proj 𝐵 )
5	3 4	uneq12i	⊢ ( pr2 𝐴 ∪ pr2 𝐵 ) = ( ( 1o Proj 𝐴 ) ∪ ( 1o Proj 𝐵 ) )
6	1 2 5	3eqtr4i	⊢ pr2 ( 𝐴 ∪ 𝐵 ) = ( pr2 𝐴 ∪ pr2 𝐵 )