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Theorem cnvcnvss 5587
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 5584 . 2 𝐴 = (𝐴 ∩ (V × V))
2 inss1 3831 . 2 (𝐴 ∩ (V × V)) ⊆ 𝐴
31, 2eqsstri 3633 1 𝐴𝐴
Colors of variables: wff setvar class
Syntax hints:  Vcvv 3198  cin 3571  wss 3572   × cxp 5110  ccnv 5111
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rab 2920  df-v 3200  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-br 4652  df-opab 4711  df-xp 5118  df-rel 5119  df-cnv 5120
This theorem is referenced by:  funcnvcnv  5954  foimacnv  6152  cnvct  8030  cnvfi  8245  structcnvcnv  15865  strlemor1OLD  15963  mvdco  17859  fcoinver  29402  fcnvgreu  29457  cnvssb  37718  relnonrel  37719  clcnvlem  37756  cnvtrrel  37788  relexpaddss  37836
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