Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  br2coss Structured version   Visualization version   GIF version

Theorem br2coss 38439
Description: Cosets by 𝑅 binary relation. (Contributed by Peter Mazsa, 25-Aug-2019.)
Assertion
Ref Expression
br2coss ((𝐴𝑉𝐵𝑊) → (𝐴 ≀ ≀ 𝑅𝐵 ↔ ([𝐴] ≀ 𝑅 ∩ [𝐵] ≀ 𝑅) ≠ ∅))

Proof of Theorem br2coss
StepHypRef Expression
1 brcoss3 38434 . 2 ((𝐴𝑉𝐵𝑊) → (𝐴 ≀ ≀ 𝑅𝐵 ↔ ([𝐴]𝑅 ∩ [𝐵]𝑅) ≠ ∅))
2 cnvcosseq 38438 . . . . 5 𝑅 = ≀ 𝑅
32eceq2i 8787 . . . 4 [𝐴]𝑅 = [𝐴] ≀ 𝑅
42eceq2i 8787 . . . 4 [𝐵]𝑅 = [𝐵] ≀ 𝑅
53, 4ineq12i 4218 . . 3 ([𝐴]𝑅 ∩ [𝐵]𝑅) = ([𝐴] ≀ 𝑅 ∩ [𝐵] ≀ 𝑅)
65neeq1i 3005 . 2 (([𝐴]𝑅 ∩ [𝐵]𝑅) ≠ ∅ ↔ ([𝐴] ≀ 𝑅 ∩ [𝐵] ≀ 𝑅) ≠ ∅)
71, 6bitrdi 287 1 ((𝐴𝑉𝐵𝑊) → (𝐴 ≀ ≀ 𝑅𝐵 ↔ ([𝐴] ≀ 𝑅 ∩ [𝐵] ≀ 𝑅) ≠ ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wcel 2108  wne 2940  cin 3950  c0 4333   class class class wbr 5143  ccnv 5684  [cec 8743  ccoss 38182
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-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
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-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  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-xp 5691  df-rel 5692  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ec 8747  df-coss 38412
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator