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Theorem relres 4817
Description: A restriction is a relation. Exercise 12 of [TakeutiZaring] p. 25. (Contributed by NM, 2-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
relres  |-  Rel  ( A  |`  B )

Proof of Theorem relres
StepHypRef Expression
1 df-res 4521 . . 3  |-  ( A  |`  B )  =  ( A  i^i  ( B  X.  _V ) )
2 inss2 3267 . . 3  |-  ( A  i^i  ( B  X.  _V ) )  C_  ( B  X.  _V )
31, 2eqsstri 3099 . 2  |-  ( A  |`  B )  C_  ( B  X.  _V )
4 relxp 4618 . 2  |-  Rel  ( B  X.  _V )
5 relss 4596 . 2  |-  ( ( A  |`  B )  C_  ( B  X.  _V )  ->  ( Rel  ( B  X.  _V )  ->  Rel  ( A  |`  B ) ) )
63, 4, 5mp2 16 1  |-  Rel  ( A  |`  B )
Colors of variables: wff set class
Syntax hints:   _Vcvv 2660    i^i cin 3040    C_ wss 3041    X. cxp 4507    |` cres 4511   Rel wrel 4514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-in 3047  df-ss 3054  df-opab 3960  df-xp 4515  df-rel 4516  df-res 4521
This theorem is referenced by:  elres  4825  resiexg  4834  iss  4835  dfres2  4841  issref  4891  asymref  4894  poirr2  4901  cnvcnvres  4972  resco  5013  ressn  5049  funssres  5135  fnresdisj  5203  fnres  5209  fcnvres  5276  nfunsn  5423  fsnunfv  5589  resfunexgALT  5976  setsresg  11908
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