Theorem relres 4657

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

Step	Hyp	Ref	Expression
1		df-res 4375	. . 3 |- ( A |` B ) = ( A i^i ( B X. _V ) )
2		inss2 3187	. . 3 |- ( A i^i ( B X. _V ) ) C_ ( B X. _V )
3	1, 2	eqsstri 3029	. 2 |- ( A |` B ) C_ ( B X. _V )
4		relxp 4465	. 2 |- Rel ( B X. _V )
5		relss 4445	. 2 |- ( ( A |` B ) C_ ( B X. _V ) -> ( Rel ( B X. _V ) -> Rel ( A |` B ) ) )
6	3, 4, 5	mp2 16	1 |- Rel ( A |` B )

Syntax hints:   _Vcvv 2601   i^i cin 2972   C_ wss 2973   X. cxp 4361   |` cres 4365   Rel wrel 4368

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-in 2979  df-ss 2986  df-opab 3840  df-xp 4369  df-rel 4370  df-res 4375

This theorem is referenced by:  elres  4664  resiexg  4673  iss  4674  dfres2  4678  issref  4727  asymref  4730  poirr2  4737  cnvcnvres  4804  resco  4845  ressn  4878  funssres  4962  fnresdisj  5029  fnres  5035  fcnvres  5093  nfunsn  5228  fsnunfv  5384  resfunexgALT  5757