Theorem cnvcnvss 6196

Description: The double converse of a class is a subclass. Exercise 2 of [TakeutiZaring] p. 25. (Contributed by NM, 23-Jul-2004.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)

Assertion

Ref	Expression
cnvcnvss	⊢ ◡◡𝐴 ⊆ 𝐴

Proof of Theorem cnvcnvss

Step	Hyp	Ref	Expression
1		cnvcnv2 6195	. 2 ⊢ ◡◡𝐴 = (𝐴 ↾ V)
2		resss 6004	. 2 ⊢ (𝐴 ↾ V) ⊆ 𝐴
3	1, 2	eqsstri 3991	1 ⊢ ◡◡𝐴 ⊆ 𝐴

Syntax hints:  Vcvv 3462   ⊆ wss 3913  ^◡ccnv 5664   ↾ cres 5667

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-ext 2742  ax-sep 5262  ax-pr 5408

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2099  df-clab 2749  df-cleq 2762  df-clel 2845  df-rab 3424  df-v 3464  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-opab 5179  df-xp 5671  df-rel 5672  df-cnv 5673  df-res 5677

This theorem is referenced by:  funcnvcnv  6607  foimacnv  6842  cnvct  9034  cnvfiALT  9299  structcnvcnv  17216  mvdco  19518  fcoinver  32919  fcnvgreu  32987  cnvssb  44264  relnonrel  44265  clcnvlem  44301  cnvtrrel  44348  relexpaddss  44396  tposres3  49608