Theorem reldmress 17280

Description: The structure restriction is a proper operator, so it can be used with ovprc1 7439. (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 17279	. 2 ⊢ ↾s = (𝑤 ∈ V, 𝑎 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑎, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑎 ∩ (Base‘𝑤))⟩)))
2	1	reldmmpo 7534	1 ⊢ Rel dom ↾s

Syntax hints:  Vcvv 3457   ∩ cin 3906   ⊆ wss 3907  ifcif 4483  ⟨cop 4591  dom cdm 5651  Rel wrel 5656  ‘cfv 6525  (class class class)co 7400   sSet csts 17211  ndxcnx 17241  Basecbs 17257   ↾_s cress 17278

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5250  ax-pr 5394

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5105  df-opab 5167  df-xp 5657  df-rel 5658  df-dm 5661  df-oprab 7404  df-mpo 7405  df-ress 17279

This theorem is referenced by:  ressbas  17284  ressbasssg  17285  ressbasssOLD  17288  resseqnbas  17290  ress0  17291  ressinbas  17293  ressress  17295  wunress  17297  subcmn  19895  submomnd  20190  suborng  20945  srasca  21267  rlmsca2  21286  resstopn  23300  cphsubrglem  25293