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Theorem opelcnvg 4877
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 4043 . . 3  |-  ( x  =  A  ->  (
y R x  <->  y R A ) )
2 breq1 4042 . . 3  |-  ( y  =  B  ->  (
y R A  <->  B R A ) )
3 df-cnv 4713 . . 3  |-  `' R  =  { <. x ,  y
>.  |  y R x }
41, 2, 3brabg 4300 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A `' R B 
<->  B R A ) )
5 df-br 4040 . 2  |-  ( A `' R B  <->  <. A ,  B >.  e.  `' R
)
6 df-br 4040 . 2  |-  ( B R A  <->  <. B ,  A >.  e.  R )
74, 5, 63bitr3g 278 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( <. A ,  B >.  e.  `' R  <->  <. B ,  A >.  e.  R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1696   <.cop 3656   class class class wbr 4039   `'ccnv 4704
This theorem is referenced by:  brcnvg  4878  opelcnv  4879  fvimacnv  5656  brtpos  6259  xrlenlt  8906  elpredim  24247  brcolinear2  24753
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-cnv 4713
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