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Theorem brcnvg 5213
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 ((𝐴𝐶𝐵𝐷) → (𝐴𝑅𝐵𝐵𝑅𝐴))

Proof of Theorem brcnvg
StepHypRef Expression
1 opelcnvg 5212 . 2 ((𝐴𝐶𝐵𝐷) → (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ ⟨𝐵, 𝐴⟩ ∈ 𝑅))
2 df-br 4578 . 2 (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
3 df-br 4578 . 2 (𝐵𝑅𝐴 ↔ ⟨𝐵, 𝐴⟩ ∈ 𝑅)
41, 2, 33bitr4g 301 1 ((𝐴𝐶𝐵𝐷) → (𝐴𝑅𝐵𝐵𝑅𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  wcel 1976  cop 4130   class class class wbr 4577  ccnv 5027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-opab 4638  df-cnv 5036
This theorem is referenced by:  brcnv  5215  brelrng  5263  eliniseg  5400  relbrcnvg  5410  brcodir  5421  elpredg  5597  predep  5609  dffv2  6166  ersym  7618  brdifun  7635  eqinf  8250  inflb  8255  infglb  8256  infglbb  8257  infltoreq  8268  infempty  8272  lbinf  10825  brcnvtrclfv  13538  oduleg  16901  posglbd  16919  znleval  19667  brbtwn  25497  fcoinvbr  28605  cnvordtrestixx  29093  xrge0iifiso  29115  orvcgteel  29662  inffzOLD  30674  fv1stcnv  30731  fv2ndcnv  30732  wsuclem  30823  wsuclemOLD  30824  wsuclb  30827  colineardim1  31144  gtinfOLD  31290  brnonrel  36710  ntrneifv2  37194  gte-lte  42220  gt-lt  42221
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