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Theorem cnvcnvss 5127
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 5125 . 2  |-  `' `' A  =  ( A  i^i  ( _V  X.  _V ) )
2 inss1 3390 . 2  |-  ( A  i^i  ( _V  X.  _V ) )  C_  A
31, 2eqsstri 3209 1  |-  `' `' A  C_  A
Colors of variables: wff set class
Syntax hints:   _Vcvv 2789    i^i cin 3152    C_ wss 3153    X. cxp 4686   `'ccnv 4687
This theorem is referenced by:  funcnvcnv  5273  foimacnv  5455  cnvfi  7135  structcnvcnv  13153  strlemor1  13229  mvdco  26787
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-br 4025  df-opab 4079  df-xp 4694  df-rel 4695  df-cnv 4696
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