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Theorem resdm 5143
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 3327 . 2  |-  dom  A  C_ 
dom  A
2 relssres 5142 . 2  |-  ( ( Rel  A  /\  dom  A 
C_  dom  A )  ->  ( A  |`  dom  A
)  =  A )
31, 2mpan2 653 1  |-  ( Rel 
A  ->  ( A  |` 
dom  A )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    C_ wss 3280   dom cdm 4837    |` cres 4839   Rel wrel 4842
This theorem is referenced by:  resdm2  5319  relresfld  5355  relcoi1  5357  fnex  5920  dftpos2  6455  tfrlem11  6608  tfrlem15  6612  tfrlem16  6613  pmresg  7000  domss2  7225  axdc3lem4  8289  gruima  8633  funsseq  25339  seff  27406  sblpnf  27407  resisresindm  27957  bnj1321  29102
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-xp 4843  df-rel 4844  df-dm 4847  df-res 4849
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