Theorem bj-pr1un 37527

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 37518	. 2 ⊢ (∅ Proj (𝐴 ∪ 𝐵)) = ((∅ Proj 𝐴) ∪ (∅ Proj 𝐵))
2		df-bj-pr1 37525	. 2 ⊢ pr1 (𝐴 ∪ 𝐵) = (∅ Proj (𝐴 ∪ 𝐵))
3		df-bj-pr1 37525	. . 3 ⊢ pr1 𝐴 = (∅ Proj 𝐴)
4		df-bj-pr1 37525	. . 3 ⊢ pr1 𝐵 = (∅ Proj 𝐵)
5	3, 4	uneq12i 4128	. 2 ⊢ (pr1 𝐴 ∪ pr1 𝐵) = ((∅ Proj 𝐴) ∪ (∅ Proj 𝐵))
6	1, 2, 5	3eqtr4i 2802	1 ⊢ pr1 (𝐴 ∪ 𝐵) = (pr1 𝐴 ∪ pr1 𝐵)

Syntax hints:   = wceq 1567   ∪ cun 3911  ∅c0 4294   Proj bj-cproj 37514  pr₁ bj-cpr1 37524

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-12 2219  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-opab 5178  df-cnv 5670  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-bj-proj 37515  df-bj-pr1 37525

This theorem is referenced by:  bj-pr21val  37537