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Theorem reldmress 15753
 Description: The structure restriction is a proper operator, so it can be used with ovprc1 6582. (Contributed by Stefan O'Rear, 29-Nov-2014.)
Assertion
Ref Expression
reldmress Rel dom ↾s

Proof of Theorem reldmress
Dummy variables 𝑤 𝑎 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ress 15702 . 2 s = (𝑤 ∈ V, 𝑎 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑎, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑎 ∩ (Base‘𝑤))⟩)))
21reldmmpt2 6669 1 Rel dom ↾s
 Colors of variables: wff setvar class Syntax hints:  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  ifcif 4036  ⟨cop 4131  dom cdm 5038  Rel wrel 5043  ‘cfv 5804  (class class class)co 6549  ndxcnx 15692   sSet csts 15693  Basecbs 15695   ↾s cress 15696 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-dm 5048  df-oprab 6553  df-mpt2 6554  df-ress 15702 This theorem is referenced by:  ressbas  15757  ressbasss  15759  resslem  15760  ress0  15761  ressinbas  15763  ressress  15765  wunress  15767  subcmn  18065  srasca  19002  rlmsca2  19022  resstopn  20800  cphsubrglem  22785  submomnd  29041  suborng  29146
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