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Theorem reldmress 16542
 Description: The structure restriction is a proper operator, so it can be used with ovprc1 7187. (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 16483 . 2 s = (𝑤 ∈ V, 𝑎 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑎, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑎 ∩ (Base‘𝑤))⟩)))
21reldmmpo 7277 1 Rel dom ↾s
 Colors of variables: wff setvar class Syntax hints:  Vcvv 3493   ∩ cin 3933   ⊆ wss 3934  ifcif 4465  ⟨cop 4565  dom cdm 5548  Rel wrel 5553  ‘cfv 6348  (class class class)co 7148  ndxcnx 16472   sSet csts 16473  Basecbs 16475   ↾s cress 16476 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-rab 3145  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-dm 5558  df-oprab 7152  df-mpo 7153  df-ress 16483 This theorem is referenced by:  ressbas  16546  ressbasss  16548  resslem  16549  ress0  16550  ressinbas  16552  ressress  16554  wunress  16556  subcmn  18949  srasca  19945  rlmsca2  19965  resstopn  21786  cphsubrglem  23773  submomnd  30704  suborng  30881
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