Theorem bj-pr2un 36354

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

Assertion

Ref	Expression
bj-pr2un	⊢ pr2 (𝐴 ∪ 𝐵) = (pr2 𝐴 ∪ pr2 𝐵)

Proof of Theorem bj-pr2un

Step	Hyp	Ref	Expression
1		bj-projun 36331	. 2 ⊢ (1o Proj (𝐴 ∪ 𝐵)) = ((1o Proj 𝐴) ∪ (1o Proj 𝐵))
2		df-bj-pr2 36352	. 2 ⊢ pr2 (𝐴 ∪ 𝐵) = (1o Proj (𝐴 ∪ 𝐵))
3		df-bj-pr2 36352	. . 3 ⊢ pr2 𝐴 = (1o Proj 𝐴)
4		df-bj-pr2 36352	. . 3 ⊢ pr2 𝐵 = (1o Proj 𝐵)
5	3, 4	uneq12i 4153	. 2 ⊢ (pr2 𝐴 ∪ pr2 𝐵) = ((1o Proj 𝐴) ∪ (1o Proj 𝐵))
6	1, 2, 5	3eqtr4i 2762	1 ⊢ pr2 (𝐴 ∪ 𝐵) = (pr2 𝐴 ∪ pr2 𝐵)

Syntax hints:   = wceq 1533   ∪ cun 3938  1_oc1o 8454   Proj bj-cproj 36327  pr₂ bj-cpr2 36351

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2163  ax-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-br 5139  df-opab 5201  df-cnv 5674  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-bj-proj 36328  df-bj-pr2 36352

This theorem is referenced by:  bj-pr22val  36356