MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  resdm2 Structured version   Visualization version   GIF version

Theorem resdm2 6251
Description: A class restricted to its domain equals its double converse. (Contributed by NM, 8-Apr-2007.)
Assertion
Ref Expression
resdm2 (𝐴 ↾ dom 𝐴) = 𝐴

Proof of Theorem resdm2
StepHypRef Expression
1 rescnvcnv 6224 . 2 (𝐴 ↾ dom 𝐴) = (𝐴 ↾ dom 𝐴)
2 relcnv 6122 . . 3 Rel 𝐴
3 resdm 6044 . . 3 (Rel 𝐴 → (𝐴 ↾ dom 𝐴) = 𝐴)
42, 3ax-mp 5 . 2 (𝐴 ↾ dom 𝐴) = 𝐴
5 dmcnvcnv 5944 . . 3 dom 𝐴 = dom 𝐴
65reseq2i 5994 . 2 (𝐴 ↾ dom 𝐴) = (𝐴 ↾ dom 𝐴)
71, 4, 63eqtr3ri 2774 1 (𝐴 ↾ dom 𝐴) = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  ccnv 5684  dom cdm 5685  cres 5687  Rel wrel 5690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697
This theorem is referenced by:  resdmres  6252  fimacnvinrn  7091  dfrel5  38347
  Copyright terms: Public domain W3C validator