Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-pr2un Structured version   Visualization version   GIF version

Theorem bj-pr2un 37018
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 36995 . 2 (1o Proj (𝐴𝐵)) = ((1o Proj 𝐴) ∪ (1o Proj 𝐵))
2 df-bj-pr2 37016 . 2 pr2 (𝐴𝐵) = (1o Proj (𝐴𝐵))
3 df-bj-pr2 37016 . . 3 pr2 𝐴 = (1o Proj 𝐴)
4 df-bj-pr2 37016 . . 3 pr2 𝐵 = (1o Proj 𝐵)
53, 4uneq12i 4166 . 2 (pr2 𝐴 ∪ pr2 𝐵) = ((1o Proj 𝐴) ∪ (1o Proj 𝐵))
61, 2, 53eqtr4i 2775 1 pr2 (𝐴𝐵) = (pr2 𝐴 ∪ pr2 𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  cun 3949  1oc1o 8499   Proj bj-cproj 36991  pr2 bj-cpr2 37015
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-bj-proj 36992  df-bj-pr2 37016
This theorem is referenced by:  bj-pr22val  37020
  Copyright terms: Public domain W3C validator