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Theorem resdm2 4811
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 4783 . 2 (𝐴 ↾ dom 𝐴) = (𝐴 ↾ dom 𝐴)
2 relcnv 4703 . . 3 Rel 𝐴
3 resdm 4649 . . 3 (Rel 𝐴 → (𝐴 ↾ dom 𝐴) = 𝐴)
42, 3ax-mp 7 . 2 (𝐴 ↾ dom 𝐴) = 𝐴
5 dmcnvcnv 4558 . . 3 dom 𝐴 = dom 𝐴
65reseq2i 4609 . 2 (𝐴 ↾ dom 𝐴) = (𝐴 ↾ dom 𝐴)
71, 4, 63eqtr3ri 2069 1 (𝐴 ↾ dom 𝐴) = 𝐴
Colors of variables: wff set class
Syntax hints:   = wceq 1243  ccnv 4344  dom cdm 4345  cres 4347  Rel wrel 4350
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353  df-dm 4355  df-rn 4356  df-res 4357
This theorem is referenced by:  resdmres  4812
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