Theorem reldmress 17177

Description: The structure restriction is a proper operator, so it can be used with ovprc1 7450. (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 17176	. 2 ⊢ ↾s = (𝑤 ∈ V, 𝑎 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑎, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑎 ∩ (Base‘𝑤))⟩)))
2	1	reldmmpo 7545	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 7411   sSet csts 17098  ndxcnx 17128  Basecbs 17146   ↾_s cress 17175

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 7415  df-mpo 7416  df-ress 17176

This theorem is referenced by:  ressbas  17181  ressbasOLD  17182  ressbasssg  17183  ressbasssOLD  17186  resseqnbas  17188  resslemOLD  17189  ress0  17190  ressinbas  17192  ressress  17195  wunress  17197  wunressOLD  17198  subcmn  19707  srasca  20804  srascaOLD  20805  rlmsca2  20829  resstopn  22697  cphsubrglem  24701  submomnd  32269  suborng  32474