MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  reldmress Unicode version

Theorem reldmress 13194
Description: The structure restriction is a proper operator, so it can be used with ovprc1 5886. (Contributed by Stefan O'Rear, 29-Nov-2014.)
Assertion
Ref Expression
reldmress  |-  Rel  doms

Proof of Theorem reldmress
Dummy variables  w  a are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ress 13155 . 2  |-s  =  ( w  e.  _V ,  a  e. 
_V  |->  if ( (
Base `  w )  C_  a ,  w ,  ( w sSet  <. ( Base `  ndx ) ,  ( a  i^i  ( Base `  w ) )
>. ) ) )
21reldmmpt2 5955 1  |-  Rel  doms
Colors of variables: wff set class
Syntax hints:   _Vcvv 2788    i^i cin 3151    C_ wss 3152   ifcif 3565   <.cop 3643   dom cdm 4689   Rel wrel 4694   ` cfv 5255  (class class class)co 5858   ndxcnx 13145   sSet csts 13146   Basecbs 13148   ↾s cress 13149
This theorem is referenced by:  ressbas  13198  ressbasss  13200  resslem  13201  ress0  13202  ressinbas  13204  ressress  13205  wunress  13207  subcmn  15133  srasca  15934  rlmsca2  15953  resstopn  16916  cphsubrglem  18613
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-dm 4699  df-oprab 5862  df-mpt2 5863  df-ress 13155
  Copyright terms: Public domain W3C validator