Theorem reldmress 16542

Description: The structure restriction is a proper operator, so it can be used with ovprc1 7174. (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 16483	. 2 ⊢ ↾s = (𝑤 ∈ V, 𝑎 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑎, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑎 ∩ (Base‘𝑤))⟩)))
2	1	reldmmpo 7264	1 ⊢ Rel dom ↾s

Syntax hints:  Vcvv 3441   ∩ cin 3880   ⊆ wss 3881  ifcif 4425  ⟨cop 4531  dom cdm 5519  Rel wrel 5524  ‘cfv 6324  (class class class)co 7135  ndxcnx 16472   sSet csts 16473  Basecbs 16475   ↾_s cress 16476

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-xp 5525  df-rel 5526  df-dm 5529  df-oprab 7139  df-mpo 7140  df-ress 16483

This theorem is referenced by:  ressbas  16546  ressbasss  16548  resslem  16549  ress0  16550  ressinbas  16552  ressress  16554  wunress  16556  subcmn  18950  srasca  19946  rlmsca2  19966  resstopn  21791  cphsubrglem  23782  submomnd  30761  suborng  30939