Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dfcoss3 Structured version   Visualization version   GIF version

Theorem dfcoss3 37272
Description: Alternate definition of the class of cosets by 𝑅 (see the comment of df-coss 37269). (Contributed by Peter Mazsa, 27-Dec-2018.)
Assertion
Ref Expression
dfcoss3 𝑅 = (𝑅𝑅)

Proof of Theorem dfcoss3
Dummy variables 𝑢 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 brcnvg 5877 . . . . . 6 ((𝑥 ∈ V ∧ 𝑢 ∈ V) → (𝑥𝑅𝑢𝑢𝑅𝑥))
21el2v 3482 . . . . 5 (𝑥𝑅𝑢𝑢𝑅𝑥)
32anbi1i 624 . . . 4 ((𝑥𝑅𝑢𝑢𝑅𝑦) ↔ (𝑢𝑅𝑥𝑢𝑅𝑦))
43exbii 1850 . . 3 (∃𝑢(𝑥𝑅𝑢𝑢𝑅𝑦) ↔ ∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦))
54opabbii 5214 . 2 {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑥𝑅𝑢𝑢𝑅𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦)}
6 df-co 5684 . 2 (𝑅𝑅) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑥𝑅𝑢𝑢𝑅𝑦)}
7 df-coss 37269 . 2 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦)}
85, 6, 73eqtr4ri 2771 1 𝑅 = (𝑅𝑅)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396   = wceq 1541  wex 1781  Vcvv 3474   class class class wbr 5147  {copab 5209  ccnv 5674  ccom 5679  ccoss 37031
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  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-co 5684  df-coss 37269
This theorem is referenced by:  cossex  37277  dmcoss3  37311  funALTVfun  37556
  Copyright terms: Public domain W3C validator