Theorem relres 4855

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 4559	. . 3 |- ( A |` B ) = ( A i^i ( B X. _V ) )
2		inss2 3302	. . 3 |- ( A i^i ( B X. _V ) ) C_ ( B X. _V )
3	1, 2	eqsstri 3134	. 2 |- ( A |` B ) C_ ( B X. _V )
4		relxp 4656	. 2 |- Rel ( B X. _V )
5		relss 4634	. 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 2689   i^i cin 3075   C_ wss 3076   X. cxp 4545   |` cres 4549   Rel wrel 4552

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-in 3082  df-ss 3089  df-opab 3998  df-xp 4553  df-rel 4554  df-res 4559

This theorem is referenced by:  elres  4863  resiexg  4872  iss  4873  dfres2  4879  issref  4929  asymref  4932  poirr2  4939  cnvcnvres  5010  resco  5051  ressn  5087  funssres  5173  fnresdisj  5241  fnres  5247  fcnvres  5314  nfunsn  5463  fsnunfv  5629  resfunexgALT  6016  setsresg  12036