Theorem bj-pr2un 37673

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 37650	. 2 ⊢ (1o Proj (𝐴 ∪ 𝐵)) = ((1o Proj 𝐴) ∪ (1o Proj 𝐵))
2		df-bj-pr2 37671	. 2 ⊢ pr2 (𝐴 ∪ 𝐵) = (1o Proj (𝐴 ∪ 𝐵))
3		df-bj-pr2 37671	. . 3 ⊢ pr2 𝐴 = (1o Proj 𝐴)
4		df-bj-pr2 37671	. . 3 ⊢ pr2 𝐵 = (1o Proj 𝐵)
5	3, 4	uneq12i 4120	. 2 ⊢ (pr2 𝐴 ∪ pr2 𝐵) = ((1o Proj 𝐴) ∪ (1o Proj 𝐵))
6	1, 2, 5	3eqtr4i 2796	1 ⊢ pr2 (𝐴 ∪ 𝐵) = (pr2 𝐴 ∪ pr2 𝐵)

Syntax hints:   = wceq 1570   ∪ cun 3903  1_oc1o 8442   Proj bj-cproj 37646  pr₂ bj-cpr2 37670

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-12 2213  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-cnv 5669  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-bj-proj 37647  df-bj-pr2 37671

This theorem is referenced by:  bj-pr22val  37675