Theorem cnvcnvss 6193

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 6191	. 2 ⊢ ◡◡𝐴 = (𝐴 ∩ (V × V))
2		inss1 4228	. 2 ⊢ (𝐴 ∩ (V × V)) ⊆ 𝐴
3	1, 2	eqsstri 4016	1 ⊢ ◡◡𝐴 ⊆ 𝐴

Syntax hints:  Vcvv 3474   ∩ cin 3947   ⊆ wss 3948   × cxp 5674  ^◡ccnv 5675

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-cnv 5684

This theorem is referenced by:  funcnvcnv  6615  foimacnv  6850  cnvct  9033  cnvfiALT  9333  structcnvcnv  17085  mvdco  19312  fcoinver  31830  fcnvgreu  31893  cnvssb  42327  relnonrel  42328  clcnvlem  42364  cnvtrrel  42411  relexpaddss  42459