Theorem cnvcnvss 6191

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 6190	. 2 ⊢ ◡◡𝐴 = (𝐴 ↾ V)
2		resss 5998	. 2 ⊢ (𝐴 ↾ V) ⊆ 𝐴
3	1, 2	eqsstri 3980	1 ⊢ ◡◡𝐴 ⊆ 𝐴

This proof depends on syntax axioms:  Vcvv 3453   ⊆ wss 3902  ^◡ccnv 5658   ↾ cres 5661

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-sep 5255  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-xp 5665  df-rel 5666  df-cnv 5667  df-res 5671

This theorem is used by:  relcnvtrg  6267  funcnvcnv  6604  foimacnv  6839  cnvct  9044  cnvfiALT  9309  structcnvcnv  17249  mvdco  19573  fcoinver  33064  fcnvgreu  33132  cnvssb  44413  relnonrel  44414  clcnvlem  44450  cnvtrrel  44497  relexpaddss  44545  tposres3  49794