Theorem reldmress 16550

Description: The structure restriction is a proper operator, so it can be used with ovprc1 7195. (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 16491	. 2 ⊢ ↾s = (𝑤 ∈ V, 𝑎 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑎, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑎 ∩ (Base‘𝑤))⟩)))
2	1	reldmmpo 7285	1 ⊢ Rel dom ↾s

Syntax hints:  Vcvv 3494   ∩ cin 3935   ⊆ wss 3936  ifcif 4467  ⟨cop 4573  dom cdm 5555  Rel wrel 5560  ‘cfv 6355  (class class class)co 7156  ndxcnx 16480   sSet csts 16481  Basecbs 16483   ↾_s cress 16484

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-br 5067  df-opab 5129  df-xp 5561  df-rel 5562  df-dm 5565  df-oprab 7160  df-mpo 7161  df-ress 16491

This theorem is referenced by:  ressbas  16554  ressbasss  16556  resslem  16557  ress0  16558  ressinbas  16560  ressress  16562  wunress  16564  subcmn  18957  srasca  19953  rlmsca2  19973  resstopn  21794  cphsubrglem  23781  submomnd  30711  suborng  30888