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

Theorem dmcnvcnv 5791
 Description: The domain of the double converse of a class is equal to its domain (even when that class in not a relation, in which case dfrel2 6035 gives another proof). (Contributed by NM, 8-Apr-2007.)
Assertion
Ref Expression
dmcnvcnv dom 𝐴 = dom 𝐴

Proof of Theorem dmcnvcnv
StepHypRef Expression
1 dfdm4 5752 . 2 dom 𝐴 = ran 𝐴
2 df-rn 5554 . 2 ran 𝐴 = dom 𝐴
31, 2eqtr2i 2848 1 dom 𝐴 = dom 𝐴
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538  ◡ccnv 5542  dom cdm 5543  ran crn 5544 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 5190  ax-nul 5197  ax-pr 5318 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-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-br 5054  df-opab 5116  df-cnv 5551  df-dm 5553  df-rn 5554 This theorem is referenced by:  resdm2  6077  f1cnvcnv  6577  trrelsuperrel2dg  40316
 Copyright terms: Public domain W3C validator