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Theorem resdm 5313
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 3523 . 2  |-  dom  A  C_ 
dom  A
2 relssres 5309 . 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 1379    C_ wss 3476   dom cdm 4999    |` cres 5001   Rel wrel 5004
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-xp 5005  df-rel 5006  df-dm 5009  df-res 5011
This theorem is referenced by:  resindm  5316  resdm2  5495  relresfld  5532  relcoi1  5534  fnex  6125  dftpos2  6969  tfrlem11  7054  tfrlem15  7058  tfrlem16  7059  pmresg  7443  domss2  7673  axdc3lem4  8829  gruima  9176  funsseq  28776  seff  30792  sblpnf  30793  resisresindm  31774  bnj1321  33162
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