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Theorem opelcnvg 5126
Description: Ordered-pair membership in converse. (Contributed by NM, 13-May-1999.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
opelcnvg  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( <. A ,  B >.  e.  `' R  <->  <. B ,  A >.  e.  R ) )

Proof of Theorem opelcnvg
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq2 4403 . . 3  |-  ( x  =  A  ->  (
y R x  <->  y R A ) )
2 breq1 4402 . . 3  |-  ( y  =  B  ->  (
y R A  <->  B R A ) )
3 df-cnv 4955 . . 3  |-  `' R  =  { <. x ,  y
>.  |  y R x }
41, 2, 3brabg 4715 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A `' R B 
<->  B R A ) )
5 df-br 4400 . 2  |-  ( A `' R B  <->  <. A ,  B >.  e.  `' R
)
6 df-br 4400 . 2  |-  ( B R A  <->  <. B ,  A >.  e.  R )
74, 5, 63bitr3g 287 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( <. A ,  B >.  e.  `' R  <->  <. B ,  A >.  e.  R ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1758   <.cop 3990   class class class wbr 4399   `'ccnv 4946
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pr 4638
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-rab 2807  df-v 3078  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-sn 3985  df-pr 3987  df-op 3991  df-br 4400  df-opab 4458  df-cnv 4955
This theorem is referenced by:  brcnvg  5127  opelcnv  5128  fvimacnv  5926  brtpos  6863  xrlenlt  9552  elpredim  27780  brcolinear2  28232
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