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Theorem dfcoss3 37740
Description: Alternate definition of the class of cosets by 𝑅 (see the comment of df-coss 37737). (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 5869 . . . . . 6 ((𝑥 ∈ V ∧ 𝑢 ∈ V) → (𝑥𝑅𝑢𝑢𝑅𝑥))
21el2v 3474 . . . . 5 (𝑥𝑅𝑢𝑢𝑅𝑥)
32anbi1i 623 . . . 4 ((𝑥𝑅𝑢𝑢𝑅𝑦) ↔ (𝑢𝑅𝑥𝑢𝑅𝑦))
43exbii 1842 . . 3 (∃𝑢(𝑥𝑅𝑢𝑢𝑅𝑦) ↔ ∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦))
54opabbii 5205 . 2 {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑥𝑅𝑢𝑢𝑅𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦)}
6 df-co 5675 . 2 (𝑅𝑅) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑥𝑅𝑢𝑢𝑅𝑦)}
7 df-coss 37737 . 2 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦)}
85, 6, 73eqtr4ri 2763 1 𝑅 = (𝑅𝑅)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395   = wceq 1533  wex 1773  Vcvv 3466   class class class wbr 5138  {copab 5200  ccnv 5665  ccom 5670  ccoss 37499
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-br 5139  df-opab 5201  df-cnv 5674  df-co 5675  df-coss 37737
This theorem is referenced by:  cossex  37745  dmcoss3  37779  funALTVfun  38024
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