MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cnvcnv2 Structured version   Visualization version   GIF version

Theorem cnvcnv2 6031
Description: The double converse of a class equals its restriction to the universe. (Contributed by NM, 8-Oct-2007.)
Assertion
Ref Expression
cnvcnv2 𝐴 = (𝐴 ↾ V)

Proof of Theorem cnvcnv2
StepHypRef Expression
1 cnvcnv 6030 . 2 𝐴 = (𝐴 ∩ (V × V))
2 df-res 5548 . 2 (𝐴 ↾ V) = (𝐴 ∩ (V × V))
31, 2eqtr4i 2850 1 𝐴 = (𝐴 ↾ V)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  Vcvv 3479  cin 3917   × cxp 5534  ccnv 5535  cres 5538
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5184  ax-nul 5191  ax-pr 5311
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-rab 3141  df-v 3481  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-br 5048  df-opab 5110  df-xp 5542  df-rel 5543  df-cnv 5544  df-res 5548
This theorem is referenced by:  dfrel3  6036  rnresv  6039  rescnvcnv  6042  cocnvcnv1  6091  cocnvcnv2  6092  strfv2d  16518  resnonrel  40124
  Copyright terms: Public domain W3C validator