Theorem reldmress 17147

Description: The structure restriction is a proper operator, so it can be used with ovprc1 7393. (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 17146	. 2 ⊢ ↾s = (𝑤 ∈ V, 𝑎 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑎, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑎 ∩ (Base‘𝑤))⟩)))
2	1	reldmmpo 7488	1 ⊢ Rel dom ↾s

Syntax hints:  Vcvv 3437   ∩ cin 3897   ⊆ wss 3898  ifcif 4476  ⟨cop 4583  dom cdm 5621  Rel wrel 5626  ‘cfv 6488  (class class class)co 7354   sSet csts 17078  ndxcnx 17108  Basecbs 17124   ↾_s cress 17145

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5096  df-opab 5158  df-xp 5627  df-rel 5628  df-dm 5631  df-oprab 7358  df-mpo 7359  df-ress 17146

This theorem is referenced by:  ressbas  17151  ressbasssg  17152  ressbasssOLD  17155  resseqnbas  17157  ress0  17158  ressinbas  17160  ressress  17162  wunress  17164  subcmn  19753  submomnd  20048  suborng  20795  srasca  21118  rlmsca2  21137  resstopn  23104  cphsubrglem  25107