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Theorem opelcnvg 4604
Description: Ordered-pair membership in converse. (Contributed by NM, 13-May-1999.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
opelcnvg ((𝐴𝐶𝐵𝐷) → (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ ⟨𝐵, 𝐴⟩ ∈ 𝑅))

Proof of Theorem opelcnvg
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq2 3841 . . 3 (𝑥 = 𝐴 → (𝑦𝑅𝑥𝑦𝑅𝐴))
2 breq1 3840 . . 3 (𝑦 = 𝐵 → (𝑦𝑅𝐴𝐵𝑅𝐴))
3 df-cnv 4436 . . 3 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝑅𝑥}
41, 2, 3brabg 4087 . 2 ((𝐴𝐶𝐵𝐷) → (𝐴𝑅𝐵𝐵𝑅𝐴))
5 df-br 3838 . 2 (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
6 df-br 3838 . 2 (𝐵𝑅𝐴 ↔ ⟨𝐵, 𝐴⟩ ∈ 𝑅)
74, 5, 63bitr3g 220 1 ((𝐴𝐶𝐵𝐷) → (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ ⟨𝐵, 𝐴⟩ ∈ 𝑅))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wcel 1438  cop 3444   class class class wbr 3837  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:  brcnvg  4605  opelcnv  4606  fvimacnv  5398  cnvf1olem  5971  brtposg  6001  xrlenlt  7530
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