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Theorem relres 5133
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 4849 . . 3  |-  ( A  |`  B )  =  ( A  i^i  ( B  X.  _V ) )
2 inss2 3522 . . 3  |-  ( A  i^i  ( B  X.  _V ) )  C_  ( B  X.  _V )
31, 2eqsstri 3338 . 2  |-  ( A  |`  B )  C_  ( B  X.  _V )
4 relxp 4942 . 2  |-  Rel  ( B  X.  _V )
5 relss 4922 . 2  |-  ( ( A  |`  B )  C_  ( B  X.  _V )  ->  ( Rel  ( B  X.  _V )  ->  Rel  ( A  |`  B ) ) )
63, 4, 5mp2 9 1  |-  Rel  ( A  |`  B )
Colors of variables: wff set class
Syntax hints:   _Vcvv 2916    i^i cin 3279    C_ wss 3280    X. cxp 4835    |` cres 4839   Rel wrel 4842
This theorem is referenced by:  elres  5140  resiexg  5147  iss  5148  dfres2  5152  issref  5206  asymref  5209  poirr2  5217  cnvcnvres  5292  resco  5333  ressn  5367  funssres  5452  fnresdisj  5514  fnres  5520  fresaunres2  5574  fcnvres  5579  nfunsn  5720  dffv2  5755  fsnunfv  5892  resfunexgALT  5917  domss2  7225  fidomdm  7347  setsres  13450  pospo  14385  metustidOLD  18542  metustid  18543  ovoliunlem1  19351  dvres  19751  dvres2  19752  dvlog  20495  efopnlem2  20501  h2hlm  22436  hlimcaui  22692  dmct  24059  dfpo2  25326  dfrdg2  25366  funpartfun  25696  mapfzcons1  26663  coeq0  26700  diophrw  26707  eldioph2lem1  26708  eldioph2lem2  26709  funressnfv  27859  dfdfat2  27862
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-v 2918  df-in 3287  df-ss 3294  df-opab 4227  df-xp 4843  df-rel 4844  df-res 4849
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