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Theorem cnvcnvss 5489
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 5487 . 2 𝐴 = (𝐴 ∩ (V × V))
2 inss1 3790 . 2 (𝐴 ∩ (V × V)) ⊆ 𝐴
31, 2eqsstri 3593 1 𝐴𝐴
Colors of variables: wff setvar class
Syntax hints:  Vcvv 3168  cin 3534  wss 3535   × cxp 5022  ccnv 5023
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-sep 4699  ax-nul 4708  ax-pr 4824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ral 2896  df-rex 2897  df-rab 2900  df-v 3170  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-sn 4121  df-pr 4123  df-op 4127  df-br 4574  df-opab 4634  df-xp 5030  df-rel 5031  df-cnv 5032
This theorem is referenced by:  funcnvcnv  5852  foimacnv  6048  cnvfi  8104  structcnvcnv  15648  strlemor1  15738  mvdco  17630  fcoinver  28600  fcnvgreu  28657  cnvct  28680  cnvssb  36710  relnonrel  36711  clcnvlem  36748  cnvtrrel  36780  relexpaddss  36828
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