Theorem cnvcnvss 6187

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 6185	. 2 ⊢ ◡◡𝐴 = (𝐴 ∩ (V × V))
2		inss1 4223	. 2 ⊢ (𝐴 ∩ (V × V)) ⊆ 𝐴
3	1, 2	eqsstri 4011	1 ⊢ ◡◡𝐴 ⊆ 𝐴

Syntax hints:  Vcvv 3468   ∩ cin 3942   ⊆ wss 3943   × cxp 5667  ^◡ccnv 5668

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-cnv 5677

This theorem is referenced by:  funcnvcnv  6609  foimacnv  6844  cnvct  9036  cnvfiALT  9336  structcnvcnv  17095  mvdco  19365  fcoinver  32344  fcnvgreu  32407  cnvssb  42910  relnonrel  42911  clcnvlem  42947  cnvtrrel  42994  relexpaddss  43042