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Theorem cnvcnvss 5144
Description: The double converse of a class is a subclass. Exercise 2 of [TakeutiZaring] p. 25. (Contributed by NM, 23-Jul-2004.)
Assertion
Ref Expression
cnvcnvss  |-  `' `' A  C_  A

Proof of Theorem cnvcnvss
StepHypRef Expression
1 cnvcnv 5142 . 2  |-  `' `' A  =  ( A  i^i  ( _V  X.  _V ) )
2 inss1 3402 . 2  |-  ( A  i^i  ( _V  X.  _V ) )  C_  A
31, 2eqsstri 3221 1  |-  `' `' A  C_  A
Colors of variables: wff set class
Syntax hints:   _Vcvv 2801    i^i cin 3164    C_ wss 3165    X. cxp 4703   `'ccnv 4704
This theorem is referenced by:  funcnvcnv  5324  foimacnv  5506  cnvfi  7156  structcnvcnv  13175  strlemor1  13251  cnvct  23358  mvdco  27491
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-cnv 4713
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