Theorem reldmress 16552

Description: The structure restriction is a proper operator, so it can be used with ovprc1 7197. (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 16493	. 2 ⊢ ↾s = (𝑤 ∈ V, 𝑎 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑎, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑎 ∩ (Base‘𝑤))⟩)))
2	1	reldmmpo 7287	1 ⊢ Rel dom ↾s

Syntax hints:  Vcvv 3496   ∩ cin 3937   ⊆ wss 3938  ifcif 4469  ⟨cop 4575  dom cdm 5557  Rel wrel 5562  ‘cfv 6357  (class class class)co 7158  ndxcnx 16482   sSet csts 16483  Basecbs 16485   ↾_s cress 16486

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 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332

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 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-br 5069  df-opab 5131  df-xp 5563  df-rel 5564  df-dm 5567  df-oprab 7162  df-mpo 7163  df-ress 16493

This theorem is referenced by:  ressbas  16556  ressbasss  16558  resslem  16559  ress0  16560  ressinbas  16562  ressress  16564  wunress  16566  subcmn  18959  srasca  19955  rlmsca2  19975  resstopn  21796  cphsubrglem  23783  submomnd  30713  suborng  30890