Theorem reldmress 17161

Description: The structure restriction is a proper operator, so it can be used with ovprc1 7392. (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 17160	. 2 ⊢ ↾s = (𝑤 ∈ V, 𝑎 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑎, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑎 ∩ (Base‘𝑤))⟩)))
2	1	reldmmpo 7487	1 ⊢ Rel dom ↾s

Syntax hints:  Vcvv 3438   ∩ cin 3904   ⊆ wss 3905  ifcif 4478  ⟨cop 4585  dom cdm 5623  Rel wrel 5628  ‘cfv 6486  (class class class)co 7353   sSet csts 17092  ndxcnx 17122  Basecbs 17138   ↾_s cress 17159

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5096  df-opab 5158  df-xp 5629  df-rel 5630  df-dm 5633  df-oprab 7357  df-mpo 7358  df-ress 17160

This theorem is referenced by:  ressbas  17165  ressbasssg  17166  ressbasssOLD  17169  resseqnbas  17171  ress0  17172  ressinbas  17174  ressress  17176  wunress  17178  subcmn  19734  submomnd  20029  suborng  20779  srasca  21102  rlmsca2  21121  resstopn  23089  cphsubrglem  25093