MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  brcnvg Unicode version

Theorem brcnvg 4862
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  e.  C  /\  B  e.  D )  ->  ( A `' R B 
<->  B R A ) )

Proof of Theorem brcnvg
StepHypRef Expression
1 opelcnvg 4861 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( <. A ,  B >.  e.  `' R  <->  <. B ,  A >.  e.  R ) )
2 df-br 4024 . 2  |-  ( A `' R B  <->  <. A ,  B >.  e.  `' R
)
3 df-br 4024 . 2  |-  ( B R A  <->  <. B ,  A >.  e.  R )
41, 2, 33bitr4g 279 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A `' R B 
<->  B R A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1684   <.cop 3643   class class class wbr 4023   `'ccnv 4688
This theorem is referenced by:  brcnv  4864  brelrng  4908  eliniseg  5042  relbrcnvg  5052  brcodir  5062  dffv2  5592  ersym  6672  brdifun  6687  lbinfm  9707  infmrgelb  9734  infmrlb  9735  infmxrlb  10652  infmxrgelb  10653  oduleg  14236  posglbd  14253  znleval  16508  ballotlemirc  23090  cnvordtrestixx  23297  xrge0iifiso  23317  orvcgteel  23668  inffz  24095  elpredg  24178  predep  24192  brbtwn  24527  colineardim1  24684  cnvref  25065  mnlmxl2  25269  nfwpr4c  25285  toplat  25290  gtinf  26234  infrglb  27722  gte-lte  28194  gt-lt  28195
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-cnv 4697
  Copyright terms: Public domain W3C validator