Theorem bj-pr2un 37502

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 37479	. 2 ⊢ (1o Proj (𝐴 ∪ 𝐵)) = ((1o Proj 𝐴) ∪ (1o Proj 𝐵))
2		df-bj-pr2 37500	. 2 ⊢ pr2 (𝐴 ∪ 𝐵) = (1o Proj (𝐴 ∪ 𝐵))
3		df-bj-pr2 37500	. . 3 ⊢ pr2 𝐴 = (1o Proj 𝐴)
4		df-bj-pr2 37500	. . 3 ⊢ pr2 𝐵 = (1o Proj 𝐵)
5	3, 4	uneq12i 4119	. 2 ⊢ (pr2 𝐴 ∪ pr2 𝐵) = ((1o Proj 𝐴) ∪ (1o Proj 𝐵))
6	1, 2, 5	3eqtr4i 2795	1 ⊢ pr2 (𝐴 ∪ 𝐵) = (pr2 𝐴 ∪ pr2 𝐵)

Syntax hints:   = wceq 1560   ∪ cun 3902  1_oc1o 8430   Proj bj-cproj 37475  pr₂ bj-cpr2 37499

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-12 2212  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-cnv 5655  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-bj-proj 37476  df-bj-pr2 37500

This theorem is referenced by:  bj-pr22val  37504