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Theorem cosscnv 38417
Description: Class of cosets by the converse of 𝑅 (Contributed by Peter Mazsa, 17-Jun-2020.)
Assertion
Ref Expression
cosscnv 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑥𝑅𝑢𝑦𝑅𝑢)}
Distinct variable group:   𝑢,𝑅,𝑥,𝑦

Proof of Theorem cosscnv
StepHypRef Expression
1 df-coss 38412 . 2 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦)}
2 brcnvg 5890 . . . . . 6 ((𝑢 ∈ V ∧ 𝑥 ∈ V) → (𝑢𝑅𝑥𝑥𝑅𝑢))
32el2v 3487 . . . . 5 (𝑢𝑅𝑥𝑥𝑅𝑢)
4 brcnvg 5890 . . . . . 6 ((𝑢 ∈ V ∧ 𝑦 ∈ V) → (𝑢𝑅𝑦𝑦𝑅𝑢))
54el2v 3487 . . . . 5 (𝑢𝑅𝑦𝑦𝑅𝑢)
63, 5anbi12i 628 . . . 4 ((𝑢𝑅𝑥𝑢𝑅𝑦) ↔ (𝑥𝑅𝑢𝑦𝑅𝑢))
76exbii 1848 . . 3 (∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦) ↔ ∃𝑢(𝑥𝑅𝑢𝑦𝑅𝑢))
87opabbii 5210 . 2 {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑥𝑅𝑢𝑦𝑅𝑢)}
91, 8eqtri 2765 1 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑥𝑅𝑢𝑦𝑅𝑢)}
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  wex 1779  Vcvv 3480   class class class wbr 5143  {copab 5205  ccnv 5684  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-rab 3437  df-v 3482  df-dif 3954  df-un 3956  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-cnv 5693  df-coss 38412
This theorem is referenced by: (None)
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