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Theorem cnvcnv 5040
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 4966 . . . . 5  |-  Rel  `' `' A
2 df-rel 4595 . . . . 5  |-  ( Rel  `' `' A  <->  `' `' A  C_  ( _V 
X.  _V ) )
31, 2mpbi 144 . . . 4  |-  `' `' A  C_  ( _V  X.  _V )
4 relxp 4697 . . . . 5  |-  Rel  ( _V  X.  _V )
5 dfrel2 5038 . . . . 5  |-  ( Rel  ( _V  X.  _V ) 
<->  `' `' ( _V  X.  _V )  =  ( _V  X.  _V ) )
64, 5mpbi 144 . . . 4  |-  `' `' ( _V  X.  _V )  =  ( _V  X.  _V )
73, 6sseqtrri 3163 . . 3  |-  `' `' A  C_  `' `' ( _V  X.  _V )
8 dfss 3116 . . 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 4995 . 2  |-  `' ( `' A  i^i  `' ( _V  X.  _V )
)  =  ( `' `' A  i^i  `' `' ( _V  X.  _V )
)
11 cnvin 4995 . . . 4  |-  `' ( A  i^i  ( _V 
X.  _V ) )  =  ( `' A  i^i  `' ( _V  X.  _V ) )
1211cnveqi 4763 . . 3  |-  `' `' ( A  i^i  ( _V  X.  _V ) )  =  `' ( `' A  i^i  `' ( _V  X.  _V )
)
13 inss2 3329 . . . . 5  |-  ( A  i^i  ( _V  X.  _V ) )  C_  ( _V  X.  _V )
14 df-rel 4595 . . . . 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 5038 . . . 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 2180 . 2  |-  `' ( `' A  i^i  `' ( _V  X.  _V )
)  =  ( A  i^i  ( _V  X.  _V ) )
199, 10, 183eqtr2i 2184 1  |-  `' `' A  =  ( A  i^i  ( _V  X.  _V ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1335   _Vcvv 2712    i^i cin 3101    C_ wss 3102    X. cxp 4586   `'ccnv 4587   Rel wrel 4593
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4084  ax-pow 4137  ax-pr 4171
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-br 3968  df-opab 4028  df-xp 4594  df-rel 4595  df-cnv 4596
This theorem is referenced by:  cnvcnv2  5041  cnvcnvss  5042  structcnvcnv  12276  strslfv2d  12302
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