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Theorem cnvcnvss 6150
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 6148 . 2 𝐴 = (𝐴 ∩ (V × V))
2 inss1 4192 . 2 (𝐴 ∩ (V × V)) ⊆ 𝐴
31, 2eqsstri 3982 1 𝐴𝐴
Colors of variables: wff setvar class
Syntax hints:  Vcvv 3447  cin 3913  wss 3914   × cxp 5635  ccnv 5636
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-br 5110  df-opab 5172  df-xp 5643  df-rel 5644  df-cnv 5645
This theorem is referenced by:  funcnvcnv  6572  foimacnv  6805  cnvct  8984  cnvfiALT  9284  structcnvcnv  17033  mvdco  19235  fcoinver  31578  fcnvgreu  31642  cnvssb  41950  relnonrel  41951  clcnvlem  41987  cnvtrrel  42034  relexpaddss  42082
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