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Theorem cnvcnv3 4867
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 4436 . 2 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝑅𝑥}
2 vex 2622 . . . 4 𝑦 ∈ V
3 vex 2622 . . . 4 𝑥 ∈ V
42, 3brcnv 4607 . . 3 (𝑦𝑅𝑥𝑥𝑅𝑦)
54opabbii 3897 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝑦𝑅𝑥} = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦}
61, 5eqtri 2108 1 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦}
Colors of variables: wff set class
Syntax hints:   = wceq 1289   class class class wbr 3837  {copab 3890  ccnv 4427
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-cnv 4436
This theorem is referenced by:  dfrel4v  4869
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