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Theorem br2coss 38420
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 38415 . 2 ((𝐴𝑉𝐵𝑊) → (𝐴 ≀ ≀ 𝑅𝐵 ↔ ([𝐴]𝑅 ∩ [𝐵]𝑅) ≠ ∅))
2 cnvcosseq 38419 . . . . 5 𝑅 = ≀ 𝑅
32eceq2i 8786 . . . 4 [𝐴]𝑅 = [𝐴] ≀ 𝑅
42eceq2i 8786 . . . 4 [𝐵]𝑅 = [𝐵] ≀ 𝑅
53, 4ineq12i 4226 . . 3 ([𝐴]𝑅 ∩ [𝐵]𝑅) = ([𝐴] ≀ 𝑅 ∩ [𝐵] ≀ 𝑅)
65neeq1i 3003 . 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 2106  wne 2938  cin 3962  c0 4339   class class class wbr 5148  ccnv 5688  [cec 8742  ccoss 38162
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ec 8746  df-coss 38393
This theorem is referenced by: (None)
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