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Theorem relres 4667
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 4383 . . 3  |-  ( A  |`  B )  =  ( A  i^i  ( B  X.  _V ) )
2 inss2 3194 . . 3  |-  ( A  i^i  ( B  X.  _V ) )  C_  ( B  X.  _V )
31, 2eqsstri 3030 . 2  |-  ( A  |`  B )  C_  ( B  X.  _V )
4 relxp 4475 . 2  |-  Rel  ( B  X.  _V )
5 relss 4453 . 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 2602    i^i cin 2973    C_ wss 2974    X. cxp 4369    |` cres 4373   Rel wrel 4376
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-in 2980  df-ss 2987  df-opab 3848  df-xp 4377  df-rel 4378  df-res 4383
This theorem is referenced by:  elres  4674  resiexg  4683  iss  4684  dfres2  4688  issref  4737  asymref  4740  poirr2  4747  cnvcnvres  4814  resco  4855  ressn  4888  funssres  4972  fnresdisj  5040  fnres  5046  fcnvres  5104  nfunsn  5239  fsnunfv  5395  resfunexgALT  5768
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