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Theorem cnvcosseq 38342
Description: The converse of cosets by 𝑅 are cosets by 𝑅. (Contributed by Peter Mazsa, 3-May-2019.)
Assertion
Ref Expression
cnvcosseq 𝑅 = ≀ 𝑅

Proof of Theorem cnvcosseq
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 brcosscnvcoss 38339 . . . . . 6 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥𝑅𝑦𝑦𝑅𝑥))
21el2v 3490 . . . . 5 (𝑥𝑅𝑦𝑦𝑅𝑥)
32biimpi 216 . . . 4 (𝑥𝑅𝑦𝑦𝑅𝑥)
43gen2 1794 . . 3 𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)
5 cnvsym 6143 . . 3 (𝑅 ⊆ ≀ 𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥))
64, 5mpbir 231 . 2 𝑅 ⊆ ≀ 𝑅
7 relcoss 38328 . . 3 Rel ≀ 𝑅
8 relcnveq 38227 . . 3 (Rel ≀ 𝑅 → (𝑅 ⊆ ≀ 𝑅𝑅 = ≀ 𝑅))
97, 8ax-mp 5 . 2 (𝑅 ⊆ ≀ 𝑅𝑅 = ≀ 𝑅)
106, 9mpbi 230 1 𝑅 = ≀ 𝑅
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wal 1535   = wceq 1537  Vcvv 3482  wss 3970   class class class wbr 5169  ccnv 5698  Rel wrel 5704  ccoss 38084
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-ext 2705  ax-sep 5320  ax-nul 5327  ax-pr 5450
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2712  df-cleq 2726  df-clel 2813  df-rab 3439  df-v 3484  df-dif 3973  df-un 3975  df-ss 3987  df-nul 4348  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5170  df-opab 5232  df-xp 5705  df-rel 5706  df-cnv 5707  df-coss 38316
This theorem is referenced by:  br2coss  38343
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