ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  cnvcnv Unicode version

Theorem cnvcnv 4801
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 4731 . . . . 5  |-  Rel  `' `' A
2 df-rel 4380 . . . . 5  |-  ( Rel  `' `' A  <->  `' `' A  C_  ( _V 
X.  _V ) )
31, 2mpbi 137 . . . 4  |-  `' `' A  C_  ( _V  X.  _V )
4 relxp 4475 . . . . 5  |-  Rel  ( _V  X.  _V )
5 dfrel2 4799 . . . . 5  |-  ( Rel  ( _V  X.  _V ) 
<->  `' `' ( _V  X.  _V )  =  ( _V  X.  _V ) )
64, 5mpbi 137 . . . 4  |-  `' `' ( _V  X.  _V )  =  ( _V  X.  _V )
73, 6sseqtr4i 3006 . . 3  |-  `' `' A  C_  `' `' ( _V  X.  _V )
8 dfss 2960 . . 3  |-  ( `' `' A  C_  `' `' ( _V  X.  _V )  <->  `' `' A  =  ( `' `' A  i^i  `' `' ( _V  X.  _V )
) )
97, 8mpbi 137 . 2  |-  `' `' A  =  ( `' `' A  i^i  `' `' ( _V  X.  _V )
)
10 cnvin 4759 . 2  |-  `' ( `' A  i^i  `' ( _V  X.  _V )
)  =  ( `' `' A  i^i  `' `' ( _V  X.  _V )
)
11 cnvin 4759 . . . 4  |-  `' ( A  i^i  ( _V 
X.  _V ) )  =  ( `' A  i^i  `' ( _V  X.  _V ) )
1211cnveqi 4538 . . 3  |-  `' `' ( A  i^i  ( _V  X.  _V ) )  =  `' ( `' A  i^i  `' ( _V  X.  _V )
)
13 inss2 3186 . . . . 5  |-  ( A  i^i  ( _V  X.  _V ) )  C_  ( _V  X.  _V )
14 df-rel 4380 . . . . 5  |-  ( Rel  ( A  i^i  ( _V  X.  _V ) )  <-> 
( A  i^i  ( _V  X.  _V ) ) 
C_  ( _V  X.  _V ) )
1513, 14mpbir 138 . . . 4  |-  Rel  ( A  i^i  ( _V  X.  _V ) )
16 dfrel2 4799 . . . 4  |-  ( Rel  ( A  i^i  ( _V  X.  _V ) )  <->  `' `' ( A  i^i  ( _V  X.  _V )
)  =  ( A  i^i  ( _V  X.  _V ) ) )
1715, 16mpbi 137 . . 3  |-  `' `' ( A  i^i  ( _V  X.  _V ) )  =  ( A  i^i  ( _V  X.  _V )
)
1812, 17eqtr3i 2078 . 2  |-  `' ( `' A  i^i  `' ( _V  X.  _V )
)  =  ( A  i^i  ( _V  X.  _V ) )
199, 10, 183eqtr2i 2082 1  |-  `' `' A  =  ( A  i^i  ( _V  X.  _V ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1259   _Vcvv 2574    i^i cin 2944    C_ wss 2945    X. cxp 4371   `'ccnv 4372   Rel wrel 4378
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-xp 4379  df-rel 4380  df-cnv 4381
This theorem is referenced by:  cnvcnv2  4802  cnvcnvss  4803
  Copyright terms: Public domain W3C validator