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Theorem resdm 5187
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 3369 . 2  |-  dom  A  C_ 
dom  A
2 relssres 5186 . 2  |-  ( ( Rel  A  /\  dom  A 
C_  dom  A )  ->  ( A  |`  dom  A
)  =  A )
31, 2mpan2 654 1  |-  ( Rel 
A  ->  ( A  |` 
dom  A )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    C_ wss 3322   dom cdm 4881    |` cres 4883   Rel wrel 4886
This theorem is referenced by:  resdm2  5363  relresfld  5399  relcoi1  5401  fnex  5964  dftpos2  6499  tfrlem11  6652  tfrlem15  6656  tfrlem16  6657  pmresg  7044  domss2  7269  axdc3lem4  8338  gruima  8682  funsseq  25398  seff  27529  sblpnf  27530  resisresindm  28089  bnj1321  29470
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216  df-opab 4270  df-xp 4887  df-rel 4888  df-dm 4891  df-res 4893
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