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Theorem cnvcnv 4961
Description: The double converse of a class strips out all elements that are not ordered pairs. (Contributed by NM, 8-Dec-2003.)
Assertion
Ref Expression
cnvcnv  |-  `' `' A  =  ( A  i^i  ( _V  X.  _V ) )

Proof of Theorem cnvcnv
StepHypRef Expression
1 relcnv 4887 . . . . 5  |-  Rel  `' `' A
2 df-rel 4516 . . . . 5  |-  ( Rel  `' `' A  <->  `' `' A  C_  ( _V 
X.  _V ) )
31, 2mpbi 144 . . . 4  |-  `' `' A  C_  ( _V  X.  _V )
4 relxp 4618 . . . . 5  |-  Rel  ( _V  X.  _V )
5 dfrel2 4959 . . . . 5  |-  ( Rel  ( _V  X.  _V ) 
<->  `' `' ( _V  X.  _V )  =  ( _V  X.  _V ) )
64, 5mpbi 144 . . . 4  |-  `' `' ( _V  X.  _V )  =  ( _V  X.  _V )
73, 6sseqtrri 3102 . . 3  |-  `' `' A  C_  `' `' ( _V  X.  _V )
8 dfss 3055 . . 3  |-  ( `' `' A  C_  `' `' ( _V  X.  _V )  <->  `' `' A  =  ( `' `' A  i^i  `' `' ( _V  X.  _V )
) )
97, 8mpbi 144 . 2  |-  `' `' A  =  ( `' `' A  i^i  `' `' ( _V  X.  _V )
)
10 cnvin 4916 . 2  |-  `' ( `' A  i^i  `' ( _V  X.  _V )
)  =  ( `' `' A  i^i  `' `' ( _V  X.  _V )
)
11 cnvin 4916 . . . 4  |-  `' ( A  i^i  ( _V 
X.  _V ) )  =  ( `' A  i^i  `' ( _V  X.  _V ) )
1211cnveqi 4684 . . 3  |-  `' `' ( A  i^i  ( _V  X.  _V ) )  =  `' ( `' A  i^i  `' ( _V  X.  _V )
)
13 inss2 3267 . . . . 5  |-  ( A  i^i  ( _V  X.  _V ) )  C_  ( _V  X.  _V )
14 df-rel 4516 . . . . 5  |-  ( Rel  ( A  i^i  ( _V  X.  _V ) )  <-> 
( A  i^i  ( _V  X.  _V ) ) 
C_  ( _V  X.  _V ) )
1513, 14mpbir 145 . . . 4  |-  Rel  ( A  i^i  ( _V  X.  _V ) )
16 dfrel2 4959 . . . 4  |-  ( Rel  ( A  i^i  ( _V  X.  _V ) )  <->  `' `' ( A  i^i  ( _V  X.  _V )
)  =  ( A  i^i  ( _V  X.  _V ) ) )
1715, 16mpbi 144 . . 3  |-  `' `' ( A  i^i  ( _V  X.  _V ) )  =  ( A  i^i  ( _V  X.  _V )
)
1812, 17eqtr3i 2140 . 2  |-  `' ( `' A  i^i  `' ( _V  X.  _V )
)  =  ( A  i^i  ( _V  X.  _V ) )
199, 10, 183eqtr2i 2144 1  |-  `' `' A  =  ( A  i^i  ( _V  X.  _V ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1316   _Vcvv 2660    i^i cin 3040    C_ wss 3041    X. cxp 4507   `'ccnv 4508   Rel wrel 4514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-xp 4515  df-rel 4516  df-cnv 4517
This theorem is referenced by:  cnvcnv2  4962  cnvcnvss  4963  structcnvcnv  11886  strslfv2d  11912
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