Theorem bj-pr1un 36967

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

Assertion

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

Proof of Theorem bj-pr1un

Step	Hyp	Ref	Expression
1		bj-projun 36958	. 2 ⊢ (∅ Proj (𝐴 ∪ 𝐵)) = ((∅ Proj 𝐴) ∪ (∅ Proj 𝐵))
2		df-bj-pr1 36965	. 2 ⊢ pr1 (𝐴 ∪ 𝐵) = (∅ Proj (𝐴 ∪ 𝐵))
3		df-bj-pr1 36965	. . 3 ⊢ pr1 𝐴 = (∅ Proj 𝐴)
4		df-bj-pr1 36965	. . 3 ⊢ pr1 𝐵 = (∅ Proj 𝐵)
5	3, 4	uneq12i 4141	. 2 ⊢ (pr1 𝐴 ∪ pr1 𝐵) = ((∅ Proj 𝐴) ∪ (∅ Proj 𝐵))
6	1, 2, 5	3eqtr4i 2768	1 ⊢ pr1 (𝐴 ∪ 𝐵) = (pr1 𝐴 ∪ pr1 𝐵)

Syntax hints:   = wceq 1540   ∪ cun 3924  ∅c0 4308   Proj bj-cproj 36954  pr₁ bj-cpr1 36964

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-br 5120  df-opab 5182  df-cnv 5662  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-bj-proj 36955  df-bj-pr1 36965

This theorem is referenced by:  bj-pr21val  36977