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 30826
Description: The scalar restriction is a proper operator, so it can be used with ovprc1 7184. (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 30825 . 2 v = (𝑦 ∈ V, 𝑥 ∈ V ↦ if((Base‘(Scalar‘𝑦)) ⊆ 𝑥, 𝑦, (𝑦 sSet ⟨(Scalar‘ndx), ((Scalar‘𝑦) ↾s 𝑥)⟩)))
21reldmmpo 7274 1 Rel dom ↾v
Colors of variables: wff setvar class
Syntax hints:  Vcvv 3492  wss 3933  ifcif 4463  cop 4563  dom cdm 5548  Rel wrel 5553  cfv 6348  (class class class)co 7145  ndxcnx 16468   sSet csts 16469  Basecbs 16471  s cress 16472  Scalarcsca 16556  v cresv 30824
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-dm 5558  df-oprab 7149  df-mpo 7150  df-resv 30825
This theorem is referenced by:  resvsca  30830  resvlem  30831
  Copyright terms: Public domain W3C validator