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Theorem cnvcnvss 5747
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 5744 . 2 𝐴 = (𝐴 ∩ (V × V))
2 inss1 3976 . 2 (𝐴 ∩ (V × V)) ⊆ 𝐴
31, 2eqsstri 3776 1 𝐴𝐴
Colors of variables: wff setvar class
Syntax hints:  Vcvv 3340  cin 3714  wss 3715   × cxp 5264  ccnv 5265
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-xp 5272  df-rel 5273  df-cnv 5274
This theorem is referenced by:  funcnvcnv  6117  foimacnv  6315  cnvct  8198  cnvfi  8413  structcnvcnv  16073  strlemor1OLD  16171  mvdco  18065  fcoinver  29725  fcnvgreu  29781  cnvssb  38394  relnonrel  38395  clcnvlem  38432  cnvtrrel  38464  relexpaddss  38512
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