Theorem reldmress 17193

Description: The structure restriction is a proper operator, so it can be used with ovprc1 7399. (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 17192	. 2 ⊢ ↾s = (𝑤 ∈ V, 𝑎 ∈ V ↦ if((Base‘𝑤) ⊆ 𝑎, 𝑤, (𝑤 sSet ⟨(Base‘ndx), (𝑎 ∩ (Base‘𝑤))⟩)))
2	1	reldmmpo 7494	1 ⊢ Rel dom ↾s

Syntax hints:  Vcvv 3430   ∩ cin 3889   ⊆ wss 3890  ifcif 4467  ⟨cop 4574  dom cdm 5624  Rel wrel 5629  ‘cfv 6492  (class class class)co 7360   sSet csts 17124  ndxcnx 17154  Basecbs 17170   ↾_s cress 17191

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-xp 5630  df-rel 5631  df-dm 5634  df-oprab 7364  df-mpo 7365  df-ress 17192

This theorem is referenced by:  ressbas  17197  ressbasssg  17198  ressbasssOLD  17201  resseqnbas  17203  ress0  17204  ressinbas  17206  ressress  17208  wunress  17210  subcmn  19803  submomnd  20098  suborng  20844  srasca  21167  rlmsca2  21186  resstopn  23161  cphsubrglem  25154