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 38415
Description: Alternate definition of the class of cosets by 𝑅 (see the comment of df-coss 38412). (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 5890 . . . . . 6 ((𝑥 ∈ V ∧ 𝑢 ∈ V) → (𝑥𝑅𝑢𝑢𝑅𝑥))
21el2v 3487 . . . . 5 (𝑥𝑅𝑢𝑢𝑅𝑥)
32anbi1i 624 . . . 4 ((𝑥𝑅𝑢𝑢𝑅𝑦) ↔ (𝑢𝑅𝑥𝑢𝑅𝑦))
43exbii 1848 . . 3 (∃𝑢(𝑥𝑅𝑢𝑢𝑅𝑦) ↔ ∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦))
54opabbii 5210 . 2 {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑥𝑅𝑢𝑢𝑅𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦)}
6 df-co 5694 . 2 (𝑅𝑅) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑥𝑅𝑢𝑢𝑅𝑦)}
7 df-coss 38412 . 2 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦)}
85, 6, 73eqtr4ri 2776 1 𝑅 = (𝑅𝑅)
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  ccom 5689  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-co 5694  df-coss 38412
This theorem is referenced by:  cossex  38420  dmcoss3  38454  funALTVfun  38699
  Copyright terms: Public domain W3C validator