Theorem bj-pr1un 34331

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 34322	. 2 ⊢ (∅ Proj (𝐴 ∪ 𝐵)) = ((∅ Proj 𝐴) ∪ (∅ Proj 𝐵))
2		df-bj-pr1 34329	. 2 ⊢ pr1 (𝐴 ∪ 𝐵) = (∅ Proj (𝐴 ∪ 𝐵))
3		df-bj-pr1 34329	. . 3 ⊢ pr1 𝐴 = (∅ Proj 𝐴)
4		df-bj-pr1 34329	. . 3 ⊢ pr1 𝐵 = (∅ Proj 𝐵)
5	3, 4	uneq12i 4125	. 2 ⊢ (pr1 𝐴 ∪ pr1 𝐵) = ((∅ Proj 𝐴) ∪ (∅ Proj 𝐵))
6	1, 2, 5	3eqtr4i 2854	1 ⊢ pr1 (𝐴 ∪ 𝐵) = (pr1 𝐴 ∪ pr1 𝐵)

Syntax hints:   = wceq 1537   ∪ cun 3922  ∅c0 4279   Proj bj-cproj 34318  pr₁ bj-cpr1 34328

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3488  df-dif 3927  df-un 3929  df-in 3931  df-ss 3940  df-nul 4280  df-if 4454  df-sn 4554  df-pr 4556  df-op 4560  df-br 5053  df-opab 5115  df-cnv 5549  df-dm 5551  df-rn 5552  df-res 5553  df-ima 5554  df-bj-proj 34319  df-bj-pr1 34329

This theorem is referenced by:  bj-pr21val  34341