Theorem cnvcnvss 6197

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 6196	. 2 ⊢ ◡◡𝐴 = (𝐴 ↾ V)
2		resss 6002	. 2 ⊢ (𝐴 ↾ V) ⊆ 𝐴
3	1, 2	eqsstri 3959	1 ⊢ ◡◡𝐴 ⊆ 𝐴

This proof depends on syntax axioms:  Vcvv 3454   ⊆ wss 3900  ^◡ccnv 5654   ↾ cres 5657

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-pr 5395

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-xp 5661  df-rel 5662  df-cnv 5663  df-res 5667

This theorem is used by:  cnvssb  6200  relcnvtrg  6283  funcnvcnv  6626  foimacnv  6861  cnvct  9087  cnvfiALT  9353  structcnvcnv  17357  mvdco  19686  fcoinver  33267  fcnvgreu  33335  relnonrel  44777  clcnvlem  44813  cnvtrrel  44860  relexpaddss  44908  tposres3  50197