Theorem cnvcnvss 6225

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	⊢ ◡◡𝐴 ⊆ 𝐴

Proof of Theorem cnvcnvss

Step	Hyp	Ref	Expression
1		cnvcnv 6223	. 2 ⊢ ◡◡𝐴 = (𝐴 ∩ (V × V))
2		inss1 4258	. 2 ⊢ (𝐴 ∩ (V × V)) ⊆ 𝐴
3	1, 2	eqsstri 4043	1 ⊢ ◡◡𝐴 ⊆ 𝐴

Syntax hints:  Vcvv 3488   ∩ cin 3975   ⊆ wss 3976   × cxp 5698  ^◡ccnv 5699

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-xp 5706  df-rel 5707  df-cnv 5708

This theorem is referenced by:  funcnvcnv  6645  foimacnv  6879  cnvct  9099  cnvfiALT  9407  structcnvcnv  17200  mvdco  19487  fcoinver  32626  fcnvgreu  32691  cnvssb  43548  relnonrel  43549  clcnvlem  43585  cnvtrrel  43632  relexpaddss  43680