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Theorem opelcnvg 5124
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 4398 . . 3  |-  ( x  =  A  ->  (
y R x  <->  y R A ) )
2 breq1 4397 . . 3  |-  ( y  =  B  ->  (
y R A  <->  B R A ) )
3 df-cnv 4950 . . 3  |-  `' R  =  { <. x ,  y
>.  |  y R x }
41, 2, 3brabg 4708 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A `' R B 
<->  B R A ) )
5 df-br 4395 . 2  |-  ( A `' R B  <->  <. A ,  B >.  e.  `' R
)
6 df-br 4395 . 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 367    e. wcel 1842   <.cop 3977   class class class wbr 4394   `'ccnv 4941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-br 4395  df-opab 4453  df-cnv 4950
This theorem is referenced by:  brcnvg  5125  opelcnv  5126  fvimacnv  5936  brtpos  6921  xrlenlt  9602  elpredim  29929  brcolinear2  30369
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