ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  brcnvg Structured version   GIF version

Theorem brcnvg 4459
Description: The converse of a binary relation swaps arguments. Theorem 11 of [Suppes] p. 61. (Contributed by NM, 10-Oct-2005.)
Assertion
Ref Expression
brcnvg ((A 𝐶 B 𝐷) → (A𝑅BB𝑅A))

Proof of Theorem brcnvg
StepHypRef Expression
1 opelcnvg 4458 . 2 ((A 𝐶 B 𝐷) → (⟨A, B 𝑅 ↔ ⟨B, A 𝑅))
2 df-br 3756 . 2 (A𝑅B ↔ ⟨A, B 𝑅)
3 df-br 3756 . 2 (B𝑅A ↔ ⟨B, A 𝑅)
41, 2, 33bitr4g 212 1 ((A 𝐶 B 𝐷) → (A𝑅BB𝑅A))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   wcel 1390  cop 3370   class class class wbr 3755  ccnv 4287
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-cnv 4296
This theorem is referenced by:  brcnv  4461  brelrng  4508  eliniseg  4638  relbrcnvg  4647  brcodir  4655  sefvex  5139  foeqcnvco  5373  isocnv2  5395  ersym  6054  brdifun  6069  ecidg  6106
  Copyright terms: Public domain W3C validator