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

Theorem cosscnv 37224
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 37219 . 2 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦)}
2 brcnvg 5877 . . . . . 6 ((𝑢 ∈ V ∧ 𝑥 ∈ V) → (𝑢𝑅𝑥𝑥𝑅𝑢))
32el2v 3483 . . . . 5 (𝑢𝑅𝑥𝑥𝑅𝑢)
4 brcnvg 5877 . . . . . 6 ((𝑢 ∈ V ∧ 𝑦 ∈ V) → (𝑢𝑅𝑦𝑦𝑅𝑢))
54el2v 3483 . . . . 5 (𝑢𝑅𝑦𝑦𝑅𝑢)
63, 5anbi12i 628 . . . 4 ((𝑢𝑅𝑥𝑢𝑅𝑦) ↔ (𝑥𝑅𝑢𝑦𝑅𝑢))
76exbii 1851 . . 3 (∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦) ↔ ∃𝑢(𝑥𝑅𝑢𝑦𝑅𝑢))
87opabbii 5214 . 2 {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑥𝑅𝑢𝑦𝑅𝑢)}
91, 8eqtri 2761 1 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑥𝑅𝑢𝑦𝑅𝑢)}
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397   = wceq 1542  wex 1782  Vcvv 3475   class class class wbr 5147  {copab 5209  ccnv 5674  ccoss 36981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-cnv 5683  df-coss 37219
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator