MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cnvcnvss Structured version   Visualization version   GIF version

Theorem cnvcnvss 6203
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 𝐴𝐴

Proof of Theorem cnvcnvss
StepHypRef Expression
1 cnvcnv 6201 . 2 𝐴 = (𝐴 ∩ (V × V))
2 inss1 4231 . 2 (𝐴 ∩ (V × V)) ⊆ 𝐴
31, 2eqsstri 4016 1 𝐴𝐴
Colors of variables: wff setvar class
Syntax hints:  Vcvv 3473  cin 3948  wss 3949   × cxp 5680  ccnv 5681
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 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-xp 5688  df-rel 5689  df-cnv 5690
This theorem is referenced by:  funcnvcnv  6625  foimacnv  6861  cnvct  9065  cnvfiALT  9366  structcnvcnv  17129  mvdco  19407  fcoinver  32415  fcnvgreu  32480  cnvssb  43047  relnonrel  43048  clcnvlem  43084  cnvtrrel  43131  relexpaddss  43179
  Copyright terms: Public domain W3C validator