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Theorem cnvcnvss 5806
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 5804 . 2 𝐴 = (𝐴 ∩ (V × V))
2 inss1 4029 . 2 (𝐴 ∩ (V × V)) ⊆ 𝐴
31, 2eqsstri 3832 1 𝐴𝐴
Colors of variables: wff setvar class
Syntax hints:  Vcvv 3386  cin 3769  wss 3770   × cxp 5311  ccnv 5312
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-sep 4976  ax-nul 4984  ax-pr 5098
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ral 3095  df-rab 3099  df-v 3388  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-sn 4370  df-pr 4372  df-op 4376  df-br 4845  df-opab 4907  df-xp 5319  df-rel 5320  df-cnv 5321
This theorem is referenced by:  funcnvcnv  6168  foimacnv  6374  cnvct  8273  cnvfi  8491  structcnvcnv  16197  mvdco  18176  fcoinver  29934  fcnvgreu  29989  cnvssb  38670  relnonrel  38671  clcnvlem  38708  cnvtrrel  38740  relexpaddss  38788
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