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

Theorem resdm2 5583
 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 5556 . 2 (𝐴 ↾ dom 𝐴) = (𝐴 ↾ dom 𝐴)
2 relcnv 5462 . . 3 Rel 𝐴
3 resdm 5400 . . 3 (Rel 𝐴 → (𝐴 ↾ dom 𝐴) = 𝐴)
42, 3ax-mp 5 . 2 (𝐴 ↾ dom 𝐴) = 𝐴
5 dmcnvcnv 5308 . . 3 dom 𝐴 = dom 𝐴
65reseq2i 5353 . 2 (𝐴 ↾ dom 𝐴) = (𝐴 ↾ dom 𝐴)
71, 4, 63eqtr3ri 2652 1 (𝐴 ↾ dom 𝐴) = 𝐴
 Colors of variables: wff setvar class Syntax hints:   = wceq 1480  ◡ccnv 5073  dom cdm 5074   ↾ cres 5076  Rel wrel 5079 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-opab 4674  df-xp 5080  df-rel 5081  df-cnv 5082  df-dm 5084  df-rn 5085  df-res 5086 This theorem is referenced by:  resdmres  5584  fimacnvinrn  6304
 Copyright terms: Public domain W3C validator