Theorem reldmress 15926

Description: The structure restriction is a proper operator, so it can be used with ovprc1 6684. (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 15865	. 2 ⊢ ↾s = (𝑤 ∈ V, 𝑎 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑎, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑎 ∩ (Base‘𝑤))⟩)))
2	1	reldmmpt2 6771	1 ⊢ Rel dom ↾s

Syntax hints:  Vcvv 3200   ∩ cin 3573   ⊆ wss 3574  ifcif 4086  ⟨cop 4183  dom cdm 5114  Rel wrel 5119  ‘cfv 5888  (class class class)co 6650  ndxcnx 15854   sSet csts 15855  Basecbs 15857   ↾_s cress 15858

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-dm 5124  df-oprab 6654  df-mpt2 6655  df-ress 15865

This theorem is referenced by:  ressbas  15930  ressbasss  15932  resslem  15933  ress0  15934  ressinbas  15936  ressress  15938  wunress  15940  subcmn  18242  srasca  19181  rlmsca2  19201  resstopn  20990  cphsubrglem  22977  submomnd  29710  suborng  29815