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Theorem dfcoss3 35838
 Description: Alternate definition of the class of cosets by 𝑅 (see the comment of df-coss 35835). (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 5714 . . . . . 6 ((𝑥 ∈ V ∧ 𝑢 ∈ V) → (𝑥𝑅𝑢𝑢𝑅𝑥))
21el2v 3448 . . . . 5 (𝑥𝑅𝑢𝑢𝑅𝑥)
32anbi1i 626 . . . 4 ((𝑥𝑅𝑢𝑢𝑅𝑦) ↔ (𝑢𝑅𝑥𝑢𝑅𝑦))
43exbii 1849 . . 3 (∃𝑢(𝑥𝑅𝑢𝑢𝑅𝑦) ↔ ∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦))
54opabbii 5097 . 2 {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑥𝑅𝑢𝑢𝑅𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦)}
6 df-co 5528 . 2 (𝑅𝑅) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑥𝑅𝑢𝑢𝑅𝑦)}
7 df-coss 35835 . 2 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦)}
85, 6, 73eqtr4ri 2832 1 𝑅 = (𝑅𝑅)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1538  ∃wex 1781  Vcvv 3441   class class class wbr 5030  {copab 5092  ◡ccnv 5518   ∘ ccom 5523   ≀ 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-cnv 5527  df-co 5528  df-coss 35835 This theorem is referenced by:  cossex  35840  dmcoss3  35869  funALTVfun  36107
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