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Theorem resdm 4572
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 2937 . 2 dom A ⊆ dom A
2 relssres 4571 . 2 ((Rel A dom A ⊆ dom A) → (A ↾ dom A) = A)
31, 2mpan2 403 1 (Rel A → (A ↾ dom A) = A)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1226  wss 2890  dom cdm 4268  cres 4270  Rel wrel 4273
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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-br 3735  df-opab 3789  df-xp 4274  df-rel 4275  df-dm 4278  df-res 4280
This theorem is referenced by:  resdm2  4734  relresfld  4770  relcoi1  4772  funimaexg  4905  fnex  5304  dftpos2  5794
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