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Theorem cnvcosseq 35858
 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 35855 . . . . . 6 ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥𝑅𝑦𝑦𝑅𝑥))
21el2v 3448 . . . . 5 (𝑥𝑅𝑦𝑦𝑅𝑥)
32biimpi 219 . . . 4 (𝑥𝑅𝑦𝑦𝑅𝑥)
43gen2 1798 . . 3 𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥)
5 cnvsym 5941 . . 3 (𝑅 ⊆ ≀ 𝑅 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝑦𝑅𝑥))
64, 5mpbir 234 . 2 𝑅 ⊆ ≀ 𝑅
7 relcoss 35844 . . 3 Rel ≀ 𝑅
8 relcnveq 35755 . . 3 (Rel ≀ 𝑅 → (𝑅 ⊆ ≀ 𝑅𝑅 = ≀ 𝑅))
97, 8ax-mp 5 . 2 (𝑅 ⊆ ≀ 𝑅𝑅 = ≀ 𝑅)
106, 9mpbi 233 1 𝑅 = ≀ 𝑅
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1536   = wceq 1538  Vcvv 3441   ⊆ wss 3881   class class class wbr 5030  ◡ccnv 5518  Rel wrel 5524   ≀ ccoss 35629 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-xp 5525  df-rel 5526  df-cnv 5527  df-coss 35835 This theorem is referenced by:  br2coss  35859
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