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Theorem resdm2 6230
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 6203 . 2 (𝐴 ↾ dom 𝐴) = (𝐴 ↾ dom 𝐴)
2 relcnv 6103 . . 3 Rel 𝐴
3 resdm 6026 . . 3 (Rel 𝐴 → (𝐴 ↾ dom 𝐴) = 𝐴)
42, 3ax-mp 5 . 2 (𝐴 ↾ dom 𝐴) = 𝐴
5 dmcnvcnv 5932 . . 3 dom 𝐴 = dom 𝐴
65reseq2i 5978 . 2 (𝐴 ↾ dom 𝐴) = (𝐴 ↾ dom 𝐴)
71, 4, 63eqtr3ri 2768 1 (𝐴 ↾ dom 𝐴) = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  ccnv 5675  dom cdm 5676  cres 5678  Rel wrel 5681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-cnv 5684  df-dm 5686  df-rn 5687  df-res 5688
This theorem is referenced by:  resdmres  6231  fimacnvinrn  7073  dfrel5  37679
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