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Theorem resdm2 6060
 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 6033 . 2 (𝐴 ↾ dom 𝐴) = (𝐴 ↾ dom 𝐴)
2 relcnv 5939 . . 3 Rel 𝐴
3 resdm 5868 . . 3 (Rel 𝐴 → (𝐴 ↾ dom 𝐴) = 𝐴)
42, 3ax-mp 5 . 2 (𝐴 ↾ dom 𝐴) = 𝐴
5 dmcnvcnv 5774 . . 3 dom 𝐴 = dom 𝐴
65reseq2i 5820 . 2 (𝐴 ↾ dom 𝐴) = (𝐴 ↾ dom 𝐴)
71, 4, 63eqtr3ri 2790 1 (𝐴 ↾ dom 𝐴) = 𝐴
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538  ◡ccnv 5523  dom cdm 5524   ↾ cres 5526  Rel wrel 5529 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-br 5033  df-opab 5095  df-xp 5530  df-rel 5531  df-cnv 5532  df-dm 5534  df-rn 5535  df-res 5536 This theorem is referenced by:  resdmres  6061  fimacnvinrn  6831  dfrel5  36043
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