Theorem reldmress 17276

Description: The structure restriction is a proper operator, so it can be used with ovprc1 7470. (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.

Step	Hyp	Ref	Expression
1		df-ress 17275	. 2 ⊢ ↾s = (𝑤 ∈ V, 𝑎 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑎, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑎 ∩ (Base‘𝑤))⟩)))
2	1	reldmmpo 7567	1 ⊢ Rel dom ↾s

Syntax hints:  Vcvv 3478   ∩ cin 3962   ⊆ wss 3963  ifcif 4531  ⟨cop 4637  dom cdm 5689  Rel wrel 5694  ‘cfv 6563  (class class class)co 7431   sSet csts 17197  ndxcnx 17227  Basecbs 17245   ↾_s cress 17274

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-dm 5699  df-oprab 7435  df-mpo 7436  df-ress 17275

This theorem is referenced by:  ressbas  17280  ressbasOLD  17281  ressbasssg  17282  ressbasssOLD  17285  resseqnbas  17287  resslemOLD  17288  ress0  17289  ressinbas  17291  ressress  17294  wunress  17296  wunressOLD  17297  subcmn  19870  srasca  21201  srascaOLD  21202  rlmsca2  21224  resstopn  23210  cphsubrglem  25225  submomnd  33070  suborng  33325