Theorem cnvcnvss 6182

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 6181	. 2 ⊢ ◡◡𝐴 = (𝐴 ↾ V)
2		resss 5989	. 2 ⊢ (𝐴 ↾ V) ⊆ 𝐴
3	1, 2	eqsstri 3984	1 ⊢ ◡◡𝐴 ⊆ 𝐴

Syntax hints:  Vcvv 3456   ⊆ wss 3906  ^◡ccnv 5648   ↾ cres 5651

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736  ax-sep 5248  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5103  df-opab 5165  df-xp 5655  df-rel 5656  df-cnv 5657  df-res 5661

This theorem is referenced by:  funcnvcnv  6590  foimacnv  6826  cnvct  9017  cnvfiALT  9284  structcnvcnv  17191  mvdco  19487  fcoinver  32806  fcnvgreu  32876  cnvssb  44167  relnonrel  44168  clcnvlem  44204  cnvtrrel  44251  relexpaddss  44299  tposres3  49507