Mathbox for BJ < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-pr1un Structured version   Visualization version   GIF version

Theorem bj-pr1un 33322
 Description: The first projection preserves unions. (Contributed by BJ, 6-Apr-2019.)
Assertion
Ref Expression
bj-pr1un pr1 (𝐴𝐵) = (pr1 𝐴 ∪ pr1 𝐵)

Proof of Theorem bj-pr1un
StepHypRef Expression
1 bj-projun 33313 . 2 (∅ Proj (𝐴𝐵)) = ((∅ Proj 𝐴) ∪ (∅ Proj 𝐵))
2 df-bj-pr1 33320 . 2 pr1 (𝐴𝐵) = (∅ Proj (𝐴𝐵))
3 df-bj-pr1 33320 . . 3 pr1 𝐴 = (∅ Proj 𝐴)
4 df-bj-pr1 33320 . . 3 pr1 𝐵 = (∅ Proj 𝐵)
53, 4uneq12i 3916 . 2 (pr1 𝐴 ∪ pr1 𝐵) = ((∅ Proj 𝐴) ∪ (∅ Proj 𝐵))
61, 2, 53eqtr4i 2803 1 pr1 (𝐴𝐵) = (pr1 𝐴 ∪ pr1 𝐵)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1631   ∪ cun 3721  ∅c0 4063   Proj bj-cproj 33309  pr1 bj-cpr1 33319 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-opab 4847  df-cnv 5257  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-bj-proj 33310  df-bj-pr1 33320 This theorem is referenced by:  bj-pr21val  33332
 Copyright terms: Public domain W3C validator