Theorem reldmress 17309

Description: The structure restriction is a proper operator, so it can be used with ovprc1 7455. (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 17308	. 2 ⊢ ↾s = (𝑤 ∈ V, 𝑎 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑎, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑎 ∩ (Base‘𝑤))⟩)))
2	1	reldmmpo 7550	1 ⊢ Rel dom ↾s

This proof depends on syntax axioms:  Vcvv 3457   ∩ cin 3905   ⊆ wss 3906  ifcif 4489  ⟨cop 4597  dom cdm 5663  Rel wrel 5668  ‘cfv 6540  (class class class)co 7416   sSet csts 17240  ndxcnx 17270  Basecbs 17286   ↾_s cress 17307

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-xp 5669  df-rel 5670  df-dm 5673  df-oprab 7420  df-mpo 7421  df-ress 17308

This theorem is used by:  ressbas  17313  ressbasssg  17314  ressbasssOLD  17317  resseqnbas  17319  ress0  17320  ressinbas  17322  ressress  17324  wunress  17326  subcmn  19930  submomnd  20225  suborng  21008  srasca  21330  rlmsca2  21349  resstopn  23372  cphsubrglem  25365