Theorem reldmress 17258

Description: The structure restriction is a proper operator, so it can be used with ovprc1 7429. (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 17257	. 2 ⊢ ↾s = (𝑤 ∈ V, 𝑎 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑎, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑎 ∩ (Base‘𝑤))⟩)))
2	1	reldmmpo 7524	1 ⊢ Rel dom ↾s

Syntax hints:  Vcvv 3453   ∩ cin 3901   ⊆ wss 3902  ifcif 4477  ⟨cop 4585  dom cdm 5643  Rel wrel 5648  ‘cfv 6515  (class class class)co 7390   sSet csts 17189  ndxcnx 17219  Basecbs 17235   ↾_s cress 17256

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-br 5098  df-opab 5160  df-xp 5649  df-rel 5650  df-dm 5653  df-oprab 7394  df-mpo 7395  df-ress 17257

This theorem is referenced by:  ressbas  17262  ressbasssg  17263  ressbasssOLD  17266  resseqnbas  17268  ress0  17269  ressinbas  17271  ressress  17273  wunress  17275  subcmn  19867  submomnd  20162  suborng  20912  srasca  21234  rlmsca2  21253  resstopn  23233  cphsubrglem  25226