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Theorem cnvcnv 5063
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 4989 . . . . 5  |-  Rel  `' `' A
2 df-rel 4618 . . . . 5  |-  ( Rel  `' `' A  <->  `' `' A  C_  ( _V 
X.  _V ) )
31, 2mpbi 144 . . . 4  |-  `' `' A  C_  ( _V  X.  _V )
4 relxp 4720 . . . . 5  |-  Rel  ( _V  X.  _V )
5 dfrel2 5061 . . . . 5  |-  ( Rel  ( _V  X.  _V ) 
<->  `' `' ( _V  X.  _V )  =  ( _V  X.  _V ) )
64, 5mpbi 144 . . . 4  |-  `' `' ( _V  X.  _V )  =  ( _V  X.  _V )
73, 6sseqtrri 3182 . . 3  |-  `' `' A  C_  `' `' ( _V  X.  _V )
8 dfss 3135 . . 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 5018 . 2  |-  `' ( `' A  i^i  `' ( _V  X.  _V )
)  =  ( `' `' A  i^i  `' `' ( _V  X.  _V )
)
11 cnvin 5018 . . . 4  |-  `' ( A  i^i  ( _V 
X.  _V ) )  =  ( `' A  i^i  `' ( _V  X.  _V ) )
1211cnveqi 4786 . . 3  |-  `' `' ( A  i^i  ( _V  X.  _V ) )  =  `' ( `' A  i^i  `' ( _V  X.  _V )
)
13 inss2 3348 . . . . 5  |-  ( A  i^i  ( _V  X.  _V ) )  C_  ( _V  X.  _V )
14 df-rel 4618 . . . . 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 5061 . . . 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 2193 . 2  |-  `' ( `' A  i^i  `' ( _V  X.  _V )
)  =  ( A  i^i  ( _V  X.  _V ) )
199, 10, 183eqtr2i 2197 1  |-  `' `' A  =  ( A  i^i  ( _V  X.  _V ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1348   _Vcvv 2730    i^i cin 3120    C_ wss 3121    X. cxp 4609   `'ccnv 4610   Rel wrel 4616
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617  df-rel 4618  df-cnv 4619
This theorem is referenced by:  cnvcnv2  5064  cnvcnvss  5065  structcnvcnv  12432  strslfv2d  12458
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