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

Theorem resdm 4993
Description: A relation restricted to its domain equals itself. (Contributed by NM, 12-Dec-2006.)
Assertion
Ref Expression
resdm  |-  ( Rel 
A  ->  ( A  |` 
dom  A )  =  A )

Proof of Theorem resdm
StepHypRef Expression
1 ssid 3197 . 2  |-  dom  A  C_ 
dom  A
2 relssres 4992 . 2  |-  ( ( Rel  A  /\  dom  A 
C_  dom  A )  ->  ( A  |`  dom  A
)  =  A )
31, 2mpan2 652 1  |-  ( Rel 
A  ->  ( A  |` 
dom  A )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    C_ wss 3152   dom cdm 4689    |` cres 4691   Rel wrel 4694
This theorem is referenced by:  resdm2  5163  relresfld  5199  relcoi1  5201  fnex  5741  dftpos2  6251  tfrlem11  6404  tfrlem15  6408  tfrlem16  6409  pmresg  6795  domss2  7020  axdc3lem4  8079  gruima  8424  funsseq  24125  seff  27538  sblpnf  27539  bnj1321  29057
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-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-res 4701
  Copyright terms: Public domain W3C validator