Theorem bj-pr1un 37698

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 37689	. 2 ⊢ (∅ Proj (𝐴 ∪ 𝐵)) = ((∅ Proj 𝐴) ∪ (∅ Proj 𝐵))
2		df-bj-pr1 37696	. 2 ⊢ pr1 (𝐴 ∪ 𝐵) = (∅ Proj (𝐴 ∪ 𝐵))
3		df-bj-pr1 37696	. . 3 ⊢ pr1 𝐴 = (∅ Proj 𝐴)
4		df-bj-pr1 37696	. . 3 ⊢ pr1 𝐵 = (∅ Proj 𝐵)
5	3, 4	uneq12i 4120	. 2 ⊢ (pr1 𝐴 ∪ pr1 𝐵) = ((∅ Proj 𝐴) ∪ (∅ Proj 𝐵))
6	1, 2, 5	3eqtr4i 2798	1 ⊢ pr1 (𝐴 ∪ 𝐵) = (pr1 𝐴 ∪ pr1 𝐵)

This proof depends on syntax axioms:   = wceq 1570   ∪ cun 3904  ∅c0 4286   Proj bj-cproj 37685  pr₁ bj-cpr1 37695

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-12 2216  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-cnv 5671  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-bj-proj 37686  df-bj-pr1 37696

This theorem is used by:  bj-pr21val  37708