Theorem reldmress 17410

Description: The structure restriction is a proper operator, so it can be used with ovprc1 7459. (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 17409	. 2 ⊢ ↾s = (𝑤 ∈ V, 𝑎 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑎, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑎 ∩ (Base‘𝑤))⟩)))
2	1	reldmmpo 7554	1 ⊢ Rel dom ↾s

This proof depends on syntax axioms:  Vcvv 3451   ∩ cin 3898   ⊆ wss 3899  ifcif 4482  ⟨cop 4590  dom cdm 5651  Rel wrel 5656  ‘cfv 6538  (class class class)co 7420   sSet csts 17341  ndxcnx 17371  Basecbs 17387   ↾_s cress 17408

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-pr 5391

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-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-xp 5657  df-rel 5658  df-dm 5661  df-oprab 7424  df-mpo 7425  df-ress 17409

This theorem is used by:  ressbas  17414  ressbasssg  17415  ressbasssOLD  17418  resseqnbas  17420  ress0  17421  ressinbas  17423  ressress  17425  wunress  17427  subcmn  20051  submomnd  20346  suborng  21133  srasca  21455  rlmsca2  21474  resstopn  23504  cphsubrglem  25498