Theorem cnvcnvss 6183

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 6181	. 2 ⊢ ◡◡𝐴 = (𝐴 ∩ (V × V))
2		inss1 4212	. 2 ⊢ (𝐴 ∩ (V × V)) ⊆ 𝐴
3	1, 2	eqsstri 4005	1 ⊢ ◡◡𝐴 ⊆ 𝐴

Syntax hints:  Vcvv 3459   ∩ cin 3925   ⊆ wss 3926   × cxp 5652  ^◡ccnv 5653

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-br 5120  df-opab 5182  df-xp 5660  df-rel 5661  df-cnv 5662

This theorem is referenced by:  funcnvcnv  6603  foimacnv  6835  cnvct  9048  cnvfiALT  9351  structcnvcnv  17172  mvdco  19426  fcoinver  32585  fcnvgreu  32651  cnvssb  43610  relnonrel  43611  clcnvlem  43647  cnvtrrel  43694  relexpaddss  43742  tposres3  48856