Theorem bj-pr2un 34830

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 34807	. 2 ⊢ (1o Proj (𝐴 ∪ 𝐵)) = ((1o Proj 𝐴) ∪ (1o Proj 𝐵))
2		df-bj-pr2 34828	. 2 ⊢ pr2 (𝐴 ∪ 𝐵) = (1o Proj (𝐴 ∪ 𝐵))
3		df-bj-pr2 34828	. . 3 ⊢ pr2 𝐴 = (1o Proj 𝐴)
4		df-bj-pr2 34828	. . 3 ⊢ pr2 𝐵 = (1o Proj 𝐵)
5	3, 4	uneq12i 4051	. 2 ⊢ (pr2 𝐴 ∪ pr2 𝐵) = ((1o Proj 𝐴) ∪ (1o Proj 𝐵))
6	1, 2, 5	3eqtr4i 2771	1 ⊢ pr2 (𝐴 ∪ 𝐵) = (pr2 𝐴 ∪ pr2 𝐵)

Syntax hints:   = wceq 1542   ∪ cun 3841  1_oc1o 8124   Proj bj-cproj 34803  pr₂ bj-cpr2 34827

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-12 2179  ax-ext 2710

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-ex 1787  df-nf 1791  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-rab 3062  df-v 3400  df-un 3848  df-in 3850  df-ss 3860  df-sn 4517  df-pr 4519  df-op 4523  df-br 5031  df-opab 5093  df-cnv 5533  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-bj-proj 34804  df-bj-pr2 34828

This theorem is referenced by:  bj-pr22val  34832