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Theorem cnvcnvss 6198
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 𝐴𝐴

Proof of Theorem cnvcnvss
StepHypRef Expression
1 cnvcnv 6196 . 2 𝐴 = (𝐴 ∩ (V × V))
2 inss1 4229 . 2 (𝐴 ∩ (V × V)) ⊆ 𝐴
31, 2eqsstri 4014 1 𝐴𝐴
Colors of variables: wff setvar class
Syntax hints:  Vcvv 3471  cin 3946  wss 3947   × cxp 5676  ccnv 5677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-opab 5211  df-xp 5684  df-rel 5685  df-cnv 5686
This theorem is referenced by:  funcnvcnv  6620  foimacnv  6856  cnvct  9058  cnvfiALT  9358  structcnvcnv  17121  mvdco  19399  fcoinver  32393  fcnvgreu  32458  cnvssb  43016  relnonrel  43017  clcnvlem  43053  cnvtrrel  43100  relexpaddss  43148
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