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Theorem cnvcnv3 5060
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 4619 . 2 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝑅𝑥}
2 vex 2733 . . . 4 𝑦 ∈ V
3 vex 2733 . . . 4 𝑥 ∈ V
42, 3brcnv 4794 . . 3 (𝑦𝑅𝑥𝑥𝑅𝑦)
54opabbii 4056 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝑦𝑅𝑥} = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦}
61, 5eqtri 2191 1 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦}
Colors of variables: wff set class
Syntax hints:   = wceq 1348   class class class wbr 3989  {copab 4049  ccnv 4610
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-cnv 4619
This theorem is referenced by:  dfrel4v  5062
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