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Theorem relres 4936
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 4646 . . 3  |-  ( A  |`  B )  =  ( A  i^i  ( B  X.  _V ) )
2 inss2 3332 . . 3  |-  ( A  i^i  ( B  X.  _V ) )  C_  ( B  X.  _V )
31, 2eqsstri 3150 . 2  |-  ( A  |`  B )  C_  ( B  X.  _V )
4 relxp 4747 . 2  |-  Rel  ( B  X.  _V )
5 relss 4728 . 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 2740    i^i cin 3093    C_ wss 3094    X. cxp 4624    |` cres 4628   Rel wrel 4631
This theorem is referenced by:  elres  4943  resiexg  4950  iss  4951  dfres2  4955  issref  5009  asymref  5012  poirr2  5020  cnvcnvres  5088  resco  5129  ressn  5163  funssres  5197  fnresdisj  5257  fnres  5263  fresaunres2  5316  fcnvres  5321  nfunsn  5457  dffv2  5491  fsnunfv  5619  resfunexgALT  5637  domss2  6953  fidomdm  7071  setsres  13101  pospo  14034  ovoliunlem1  18788  dvres  19188  dvres2  19189  dvlog  19925  efopnlem2  19931  h2hlm  21485  hlimcaui  21741  dfpo2  23448  dfrdg2  23486  funpartfun  23821  restidsing  24407  dispos  24619  mapfzcons1  26126  coeq0  26163  diophrw  26170  eldioph2lem1  26171  eldioph2lem2  26172
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 1927  ax-ext 2237
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 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-v 2742  df-in 3101  df-ss 3108  df-opab 4018  df-xp 4640  df-rel 4641  df-res 4646
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