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Theorem cnvcnv3 5035
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 4594 . 2 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝑅𝑥}
2 vex 2715 . . . 4 𝑦 ∈ V
3 vex 2715 . . . 4 𝑥 ∈ V
42, 3brcnv 4769 . . 3 (𝑦𝑅𝑥𝑥𝑅𝑦)
54opabbii 4031 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝑦𝑅𝑥} = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦}
61, 5eqtri 2178 1 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦}
Colors of variables: wff set class
Syntax hints:   = wceq 1335   class class class wbr 3965  {copab 4024  ccnv 4585
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966  df-opab 4026  df-cnv 4594
This theorem is referenced by:  dfrel4v  5037
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