Theorem bj-pr2un 32198

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 32175	. 2 ⊢ (1𝑜 Proj (𝐴 ∪ 𝐵)) = ((1𝑜 Proj 𝐴) ∪ (1𝑜 Proj 𝐵))
2		df-bj-pr2 32196	. 2 ⊢ pr2 (𝐴 ∪ 𝐵) = (1𝑜 Proj (𝐴 ∪ 𝐵))
3		df-bj-pr2 32196	. . 3 ⊢ pr2 𝐴 = (1𝑜 Proj 𝐴)
4		df-bj-pr2 32196	. . 3 ⊢ pr2 𝐵 = (1𝑜 Proj 𝐵)
5	3, 4	uneq12i 3727	. 2 ⊢ (pr2 𝐴 ∪ pr2 𝐵) = ((1𝑜 Proj 𝐴) ∪ (1𝑜 Proj 𝐵))
6	1, 2, 5	3eqtr4i 2642	1 ⊢ pr2 (𝐴 ∪ 𝐵) = (pr2 𝐴 ∪ pr2 𝐵)

Syntax hints:   = wceq 1475   ∪ cun 3538  1_𝑜c1o 7440   Proj bj-cproj 32171  pr₂ bj-cpr2 32195

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-bj-proj 32172  df-bj-pr2 32196

This theorem is referenced by:  bj-pr22val  32200