Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  reldmresv Structured version   Visualization version   GIF version

Theorem reldmresv 30166
Description: The scalar restriction is a proper operator, so it can be used with ovprc1 6829. (Contributed by Thierry Arnoux, 6-Sep-2018.)
Assertion
Ref Expression
reldmresv Rel dom ↾v

Proof of Theorem reldmresv
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-resv 30165 . 2 v = (𝑦 ∈ V, 𝑥 ∈ V ↦ if((Base‘(Scalar‘𝑦)) ⊆ 𝑥, 𝑦, (𝑦 sSet ⟨(Scalar‘ndx), ((Scalar‘𝑦) ↾s 𝑥)⟩)))
21reldmmpt2 6918 1 Rel dom ↾v
Colors of variables: wff setvar class
Syntax hints:  Vcvv 3351  wss 3723  ifcif 4225  cop 4322  dom cdm 5249  Rel wrel 5254  cfv 6031  (class class class)co 6793  ndxcnx 16061   sSet csts 16062  Basecbs 16064  s cress 16065  Scalarcsca 16152  v cresv 30164
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-opab 4847  df-xp 5255  df-rel 5256  df-dm 5259  df-oprab 6797  df-mpt2 6798  df-resv 30165
This theorem is referenced by:  resvsca  30170  resvlem  30171
  Copyright terms: Public domain W3C validator