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Theorem cnvcnv3 6178
Description: The set of all ordered pairs in a class is the same as the double converse. (Contributed by Mario Carneiro, 16-Aug-2015.)
Assertion
Ref Expression
cnvcnv3 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦}
Distinct variable group:   𝑥,𝑦,𝑅

Proof of Theorem cnvcnv3
StepHypRef Expression
1 df-cnv 5675 . 2 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝑅𝑥}
2 vex 3470 . . . 4 𝑦 ∈ V
3 vex 3470 . . . 4 𝑥 ∈ V
42, 3brcnv 5873 . . 3 (𝑦𝑅𝑥𝑥𝑅𝑦)
54opabbii 5206 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝑦𝑅𝑥} = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦}
61, 5eqtri 2752 1 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533   class class class wbr 5139  {copab 5201  ccnv 5666
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 5290  ax-nul 5297  ax-pr 5418
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 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-cnv 5675
This theorem is referenced by:  dfrel4v  6180
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