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Theorem resdm 4592
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 2958 . 2 dom A ⊆ dom A
2 relssres 4591 . 2 ((Rel A dom A ⊆ dom A) → (A ↾ dom A) = A)
31, 2mpan2 401 1 (Rel A → (A ↾ dom A) = A)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242  wss 2911  dom cdm 4288  cres 4290  Rel wrel 4293
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-dm 4298  df-res 4300
This theorem is referenced by:  resdm2  4754  relresfld  4790  relcoi1  4792  funimaexg  4926  fnex  5326  dftpos2  5817
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