Theorem reldmress 17159

Description: The structure restriction is a proper operator, so it can be used with ovprc1 7397. (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 17158	. 2 ⊢ ↾s = (𝑤 ∈ V, 𝑎 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑎, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑎 ∩ (Base‘𝑤))⟩)))
2	1	reldmmpo 7492	1 ⊢ Rel dom ↾s

Syntax hints:  Vcvv 3440   ∩ cin 3900   ⊆ wss 3901  ifcif 4479  ⟨cop 4586  dom cdm 5624  Rel wrel 5629  ‘cfv 6492  (class class class)co 7358   sSet csts 17090  ndxcnx 17120  Basecbs 17136   ↾_s cress 17157

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 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-xp 5630  df-rel 5631  df-dm 5634  df-oprab 7362  df-mpo 7363  df-ress 17158

This theorem is referenced by:  ressbas  17163  ressbasssg  17164  ressbasssOLD  17167  resseqnbas  17169  ress0  17170  ressinbas  17172  ressress  17174  wunress  17176  subcmn  19766  submomnd  20061  suborng  20809  srasca  21132  rlmsca2  21151  resstopn  23130  cphsubrglem  25133