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Theorem opelcnvg 4860
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 ) )
Dummy variables  x  y are mutually distinct and distinct from all other variables.

Proof of Theorem opelcnvg
StepHypRef Expression
1 breq2 4028 . . 3  |-  ( x  =  A  ->  (
y R x  <->  y R A ) )
2 breq1 4027 . . 3  |-  ( y  =  B  ->  (
y R A  <->  B R A ) )
3 df-cnv 4696 . . 3  |-  `' R  =  { <. x ,  y
>.  |  y R x }
41, 2, 3brabg 4283 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A `' R B 
<->  B R A ) )
5 df-br 4025 . 2  |-  ( A `' R B  <->  <. A ,  B >.  e.  `' R
)
6 df-br 4025 . 2  |-  ( B R A  <->  <. B ,  A >.  e.  R )
74, 5, 63bitr3g 280 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( <. A ,  B >.  e.  `' R  <->  <. B ,  A >.  e.  R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    e. wcel 1685   <.cop 3644   class class class wbr 4024   `'ccnv 4687
This theorem is referenced by:  brcnvg  4861  opelcnv  4862  fvimacnv  5601  brtpos  6204  xrlenlt  8885  elpredim  23577  brcolinear2  24088
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-rab 2553  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-br 4025  df-opab 4079  df-cnv 4696
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