Theorem bj-pr2un 37000

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 36977	. 2 ⊢ (1o Proj (𝐴 ∪ 𝐵)) = ((1o Proj 𝐴) ∪ (1o Proj 𝐵))
2		df-bj-pr2 36998	. 2 ⊢ pr2 (𝐴 ∪ 𝐵) = (1o Proj (𝐴 ∪ 𝐵))
3		df-bj-pr2 36998	. . 3 ⊢ pr2 𝐴 = (1o Proj 𝐴)
4		df-bj-pr2 36998	. . 3 ⊢ pr2 𝐵 = (1o Proj 𝐵)
5	3, 4	uneq12i 4176	. 2 ⊢ (pr2 𝐴 ∪ pr2 𝐵) = ((1o Proj 𝐴) ∪ (1o Proj 𝐵))
6	1, 2, 5	3eqtr4i 2773	1 ⊢ pr2 (𝐴 ∪ 𝐵) = (pr2 𝐴 ∪ pr2 𝐵)

Syntax hints:   = wceq 1537   ∪ cun 3961  1_oc1o 8498   Proj bj-cproj 36973  pr₂ bj-cpr2 36997

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-bj-proj 36974  df-bj-pr2 36998

This theorem is referenced by:  bj-pr22val  37002