Theorem reldmress 17214

Description: The structure restriction is a proper operator, so it can be used with ovprc1 7458. (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 17213	. 2 ⊢ ↾s = (𝑤 ∈ V, 𝑎 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑎, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑎 ∩ (Base‘𝑤))⟩)))
2	1	reldmmpo 7555	1 ⊢ Rel dom ↾s

Syntax hints:  Vcvv 3461   ∩ cin 3943   ⊆ wss 3944  ifcif 4530  ⟨cop 4636  dom cdm 5678  Rel wrel 5683  ‘cfv 6549  (class class class)co 7419   sSet csts 17135  ndxcnx 17165  Basecbs 17183   ↾_s cress 17212

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5150  df-opab 5212  df-xp 5684  df-rel 5685  df-dm 5688  df-oprab 7423  df-mpo 7424  df-ress 17213

This theorem is referenced by:  ressbas  17218  ressbasOLD  17219  ressbasssg  17220  ressbasssOLD  17223  resseqnbas  17225  resslemOLD  17226  ress0  17227  ressinbas  17229  ressress  17232  wunress  17234  wunressOLD  17235  subcmn  19804  srasca  21081  srascaOLD  21082  rlmsca2  21104  resstopn  23134  cphsubrglem  25149  submomnd  32880  suborng  33129