Theorem bj-pr1un 32991

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 32982	. 2 ⊢ (∅ Proj (𝐴 ∪ 𝐵)) = ((∅ Proj 𝐴) ∪ (∅ Proj 𝐵))
2		df-bj-pr1 32989	. 2 ⊢ pr1 (𝐴 ∪ 𝐵) = (∅ Proj (𝐴 ∪ 𝐵))
3		df-bj-pr1 32989	. . 3 ⊢ pr1 𝐴 = (∅ Proj 𝐴)
4		df-bj-pr1 32989	. . 3 ⊢ pr1 𝐵 = (∅ Proj 𝐵)
5	3, 4	uneq12i 3765	. 2 ⊢ (pr1 𝐴 ∪ pr1 𝐵) = ((∅ Proj 𝐴) ∪ (∅ Proj 𝐵))
6	1, 2, 5	3eqtr4i 2654	1 ⊢ pr1 (𝐴 ∪ 𝐵) = (pr1 𝐴 ∪ pr1 𝐵)

Syntax hints:   = wceq 1483   ∪ cun 3572  ∅c0 3915   Proj bj-cproj 32978  pr₁ bj-cpr1 32988

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-bj-proj 32979  df-bj-pr1 32989

This theorem is referenced by:  bj-pr21val  33001