Metamath Proof Explorer


Theorem bj-pr1un

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

Ref Expression
Assertion bj-pr1un pr1 ( 𝐴𝐵 ) = ( pr1 𝐴 ∪ pr1 𝐵 )

Proof

Step Hyp Ref Expression
1 bj-projun ( ∅ Proj ( 𝐴𝐵 ) ) = ( ( ∅ Proj 𝐴 ) ∪ ( ∅ Proj 𝐵 ) )
2 df-bj-pr1 pr1 ( 𝐴𝐵 ) = ( ∅ Proj ( 𝐴𝐵 ) )
3 df-bj-pr1 pr1 𝐴 = ( ∅ Proj 𝐴 )
4 df-bj-pr1 pr1 𝐵 = ( ∅ Proj 𝐵 )
5 3 4 uneq12i ( pr1 𝐴 ∪ pr1 𝐵 ) = ( ( ∅ Proj 𝐴 ) ∪ ( ∅ Proj 𝐵 ) )
6 1 2 5 3eqtr4i pr1 ( 𝐴𝐵 ) = ( pr1 𝐴 ∪ pr1 𝐵 )