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Theorem opelcnvg 4768
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
StepHypRef Expression
1 breq2 3924 . . 3  |-  ( x  =  A  ->  (
y R x  <->  y R A ) )
2 breq1 3923 . . 3  |-  ( y  =  B  ->  (
y R A  <->  B R A ) )
3 df-cnv 4596 . . 3  |-  `' R  =  { <. x ,  y
>.  |  y R x }
41, 2, 3brabg 4177 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A `' R B 
<->  B R A ) )
5 df-br 3921 . 2  |-  ( A `' R B  <->  <. A ,  B >.  e.  `' R
)
6 df-br 3921 . 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 1621   <.cop 3547   class class class wbr 3920   `'ccnv 4579
This theorem is referenced by:  brcnvg  4769  opelcnv  4770  fvimacnv  5492  brtpos  6095  xrlenlt  8770  elpredim  23344  brcolinear2  23855
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-rab 2516  df-v 2729  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-br 3921  df-opab 3975  df-cnv 4596
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