Theorem reldmress 17174

Description: The structure restriction is a proper operator, so it can be used with ovprc1 7447. (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 17173	. 2 ⊢ ↾s = (𝑤 ∈ V, 𝑎 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑎, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑎 ∩ (Base‘𝑤))⟩)))
2	1	reldmmpo 7542	1 ⊢ Rel dom ↾s

Syntax hints:  Vcvv 3474   ∩ cin 3947   ⊆ wss 3948  ifcif 4528  ⟨cop 4634  dom cdm 5676  Rel wrel 5681  ‘cfv 6543  (class class class)co 7408   sSet csts 17095  ndxcnx 17125  Basecbs 17143   ↾_s cress 17172

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-dm 5686  df-oprab 7412  df-mpo 7413  df-ress 17173

This theorem is referenced by:  ressbas  17178  ressbasOLD  17179  ressbasssg  17180  ressbasssOLD  17183  resseqnbas  17185  resslemOLD  17186  ress0  17187  ressinbas  17189  ressress  17192  wunress  17194  wunressOLD  17195  subcmn  19704  srasca  20797  srascaOLD  20798  rlmsca2  20822  resstopn  22689  cphsubrglem  24693  submomnd  32223  suborng  32428