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Theorem resdm 4958
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 3187 . 2  |-  dom  A  C_ 
dom  A
2 relssres 4957 . 2  |-  ( ( Rel  A  /\  dom  A 
C_  dom  A )  ->  ( A  |`  dom  A
)  =  A )
31, 2mpan2 425 1  |-  ( Rel 
A  ->  ( A  |` 
dom  A )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1363    C_ wss 3141   dom cdm 4638    |` cres 4640   Rel wrel 4643
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-br 4016  df-opab 4077  df-xp 4644  df-rel 4645  df-dm 4648  df-res 4650
This theorem is referenced by:  resindm  4961  resdm2  5131  relresfld  5170  relcoi1  5172  funimaexg  5312  fnex  5751  dftpos2  6276  pmresg  6690  dif1en  6893
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