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Theorem cnvcnvss 6044
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 6042 . 2 𝐴 = (𝐴 ∩ (V × V))
2 inss1 4202 . 2 (𝐴 ∩ (V × V)) ⊆ 𝐴
31, 2eqsstri 3998 1 𝐴𝐴
Colors of variables: wff setvar class
Syntax hints:  Vcvv 3492  cin 3932  wss 3933   × cxp 5546  ccnv 5547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-cnv 5556
This theorem is referenced by:  funcnvcnv  6414  foimacnv  6625  cnvct  8574  cnvfi  8794  structcnvcnv  16485  mvdco  18502  fcoinver  30285  fcnvgreu  30346  cnvssb  39824  relnonrel  39825  clcnvlem  39861  cnvtrrel  39893  relexpaddss  39941
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