Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  relres Unicode version

Theorem relres 4639
 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

Proof of Theorem relres
StepHypRef Expression
1 df-res 4357 . . 3
2 inss2 3158 . . 3
31, 2eqsstri 2975 . 2
4 relxp 4447 . 2
5 relss 4427 . 2
63, 4, 5mp2 16 1
 Colors of variables: wff set class Syntax hints:  cvv 2557   cin 2916   wss 2917   cxp 4343   cres 4347   wrel 4350 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924  df-ss 2931  df-opab 3819  df-xp 4351  df-rel 4352  df-res 4357 This theorem is referenced by:  elres  4646  resiexg  4653  iss  4654  dfres2  4658  issref  4707  asymref  4710  poirr2  4717  cnvcnvres  4784  resco  4825  ressn  4858  funssres  4942  fnresdisj  5009  fnres  5015  fcnvres  5073  nfunsn  5207  fsnunfv  5363  resfunexgALT  5737
 Copyright terms: Public domain W3C validator