Description: The first projection preserves unions. (Contributed by BJ, 6-Apr-2019)
Ref | Expression | ||
---|---|---|---|
Assertion | bj-pr1un | ⊢ pr1 ( 𝐴 ∪ 𝐵 ) = ( pr1 𝐴 ∪ pr1 𝐵 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-projun | ⊢ ( ∅ Proj ( 𝐴 ∪ 𝐵 ) ) = ( ( ∅ Proj 𝐴 ) ∪ ( ∅ Proj 𝐵 ) ) | |
2 | df-bj-pr1 | ⊢ pr1 ( 𝐴 ∪ 𝐵 ) = ( ∅ Proj ( 𝐴 ∪ 𝐵 ) ) | |
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 ( 𝐴 ∪ 𝐵 ) = ( pr1 𝐴 ∪ pr1 𝐵 ) |