MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cnvcnv3 Structured version   Visualization version   GIF version

Theorem cnvcnv3 6042
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 5561 . 2 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝑅𝑥}
2 vex 3502 . . . 4 𝑦 ∈ V
3 vex 3502 . . . 4 𝑥 ∈ V
42, 3brcnv 5751 . . 3 (𝑦𝑅𝑥𝑥𝑅𝑦)
54opabbii 5129 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝑦𝑅𝑥} = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦}
61, 5eqtri 2848 1 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1530   class class class wbr 5062  {copab 5124  ccnv 5552
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pr 5325
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-rab 3151  df-v 3501  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-br 5063  df-opab 5125  df-cnv 5561
This theorem is referenced by:  dfrel4v  6044
  Copyright terms: Public domain W3C validator