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Theorem bj-pr2un 37019
Description: The second projection preserves unions. (Contributed by BJ, 6-Apr-2019.)
Assertion
Ref Expression
bj-pr2un pr2 (𝐴𝐵) = (pr2 𝐴 ∪ pr2 𝐵)

Proof of Theorem bj-pr2un
StepHypRef Expression
1 bj-projun 36996 . 2 (1o Proj (𝐴𝐵)) = ((1o Proj 𝐴) ∪ (1o Proj 𝐵))
2 df-bj-pr2 37017 . 2 pr2 (𝐴𝐵) = (1o Proj (𝐴𝐵))
3 df-bj-pr2 37017 . . 3 pr2 𝐴 = (1o Proj 𝐴)
4 df-bj-pr2 37017 . . 3 pr2 𝐵 = (1o Proj 𝐵)
53, 4uneq12i 4165 . 2 (pr2 𝐴 ∪ pr2 𝐵) = ((1o Proj 𝐴) ∪ (1o Proj 𝐵))
61, 2, 53eqtr4i 2774 1 pr2 (𝐴𝐵) = (pr2 𝐴 ∪ pr2 𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1539  cun 3948  1oc1o 8500   Proj bj-cproj 36992  pr2 bj-cpr2 37016
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-12 2176  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-cnv 5692  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-bj-proj 36993  df-bj-pr2 37017
This theorem is referenced by:  bj-pr22val  37021
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