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Theorem relres 4890
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 4600 . . 3  |-  ( A  |`  B )  =  ( A  i^i  ( B  X.  _V ) )
2 inss2 3297 . . 3  |-  ( A  i^i  ( B  X.  _V ) )  C_  ( B  X.  _V )
31, 2eqsstri 3129 . 2  |-  ( A  |`  B )  C_  ( B  X.  _V )
4 relxp 4701 . 2  |-  Rel  ( B  X.  _V )
5 relss 4682 . 2  |-  ( ( A  |`  B )  C_  ( B  X.  _V )  ->  ( Rel  ( B  X.  _V )  ->  Rel  ( A  |`  B ) ) )
63, 4, 5mp2 19 1  |-  Rel  ( A  |`  B )
Colors of variables: wff set class
Syntax hints:   _Vcvv 2727    i^i cin 3077    C_ wss 3078    X. cxp 4578    |` cres 4582   Rel wrel 4585
This theorem is referenced by:  elres  4897  resiexg  4904  iss  4905  dfres2  4909  issref  4963  asymref  4966  poirr2  4974  cnvcnvres  5042  resco  5083  ressn  5117  funssres  5151  fnresdisj  5211  fnres  5217  fresaunres2  5270  fcnvres  5275  nfunsn  5410  dffv2  5444  fsnunfv  5572  resfunexgALT  5590  domss2  6905  fidomdm  7023  setsres  13048  pospo  13951  ovoliunlem1  18693  dvres  19093  dvres2  19094  dvlog  19830  efopnlem2  19836  h2hlm  21390  hlimcaui  21646  dfpo2  23282  dfrdg2  23320  funpartfun  23655  restidsing  24241  dispos  24453  mapfzcons1  25960  coeq0  25997  diophrw  26004  eldioph2lem1  26005  eldioph2lem2  26006
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-v 2729  df-in 3085  df-ss 3089  df-opab 3975  df-xp 4594  df-rel 4595  df-res 4600
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