Theorem reldmress 16943

Description: The structure restriction is a proper operator, so it can be used with ovprc1 7314. (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 16942	. 2 ⊢ ↾s = (𝑤 ∈ V, 𝑎 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑎, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑎 ∩ (Base‘𝑤))⟩)))
2	1	reldmmpo 7408	1 ⊢ Rel dom ↾s

Syntax hints:  Vcvv 3432   ∩ cin 3886   ⊆ wss 3887  ifcif 4459  ⟨cop 4567  dom cdm 5589  Rel wrel 5594  ‘cfv 6433  (class class class)co 7275   sSet csts 16864  ndxcnx 16894  Basecbs 16912   ↾_s cress 16941

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-xp 5595  df-rel 5596  df-dm 5599  df-oprab 7279  df-mpo 7280  df-ress 16942

This theorem is referenced by:  ressbas  16947  ressbasOLD  16948  ressbasss  16950  resseqnbas  16951  resslemOLD  16952  ress0  16953  ressinbas  16955  ressress  16958  wunress  16960  wunressOLD  16961  subcmn  19438  srasca  20447  srascaOLD  20448  rlmsca2  20471  resstopn  22337  cphsubrglem  24341  submomnd  31336  suborng  31514