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Theorem resdm 5137
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 3463 . 2  |-  dom  A  C_ 
dom  A
2 relssres 5133 . 2  |-  ( ( Rel  A  /\  dom  A 
C_  dom  A )  ->  ( A  |`  dom  A
)  =  A )
31, 2mpan2 671 1  |-  ( Rel 
A  ->  ( A  |` 
dom  A )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1407    C_ wss 3416   dom cdm 4825    |` cres 4827   Rel wrel 4830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pr 4632
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-sn 3975  df-pr 3977  df-op 3981  df-br 4398  df-opab 4456  df-xp 4831  df-rel 4832  df-dm 4835  df-res 4837
This theorem is referenced by:  resindm  5140  resdm2  5315  relresfld  5352  relcoi1OLD  5355  fnex  6122  dftpos2  6977  tfrlem11  7093  tfrlem15  7097  tfrlem16  7098  pmresg  7486  domss2  7716  axdc3lem4  8867  gruima  9212  bnj1321  29423  funsseq  29995  seff  36050  sblpnf  36051  resisresindm  37951
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