Theorem reldmress 17289

Description: The structure restriction is a proper operator, so it can be used with ovprc1 7487. (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 17288	. 2 ⊢ ↾s = (𝑤 ∈ V, 𝑎 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑎, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑎 ∩ (Base‘𝑤))⟩)))
2	1	reldmmpo 7584	1 ⊢ Rel dom ↾s

Syntax hints:  Vcvv 3488   ∩ cin 3975   ⊆ wss 3976  ifcif 4548  ⟨cop 4654  dom cdm 5700  Rel wrel 5705  ‘cfv 6573  (class class class)co 7448   sSet csts 17210  ndxcnx 17240  Basecbs 17258   ↾_s cress 17287

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-xp 5706  df-rel 5707  df-dm 5710  df-oprab 7452  df-mpo 7453  df-ress 17288

This theorem is referenced by:  ressbas  17293  ressbasOLD  17294  ressbasssg  17295  ressbasssOLD  17298  resseqnbas  17300  resslemOLD  17301  ress0  17302  ressinbas  17304  ressress  17307  wunress  17309  wunressOLD  17310  subcmn  19879  srasca  21206  srascaOLD  21207  rlmsca2  21229  resstopn  23215  cphsubrglem  25230  submomnd  33060  suborng  33310