Theorem relres 4988

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 4688	. . 3 |- ( A |` B ) = ( A i^i ( B X. _V ) )
2		inss2 3394	. . 3 |- ( A i^i ( B X. _V ) ) C_ ( B X. _V )
3	1, 2	eqsstri 3225	. 2 |- ( A |` B ) C_ ( B X. _V )
4		relxp 4785	. 2 |- Rel ( B X. _V )
5		relss 4763	. 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 2772   i^i cin 3165   C_ wss 3166   X. cxp 4674   |` cres 4678   Rel wrel 4681

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-in 3172  df-ss 3179  df-opab 4107  df-xp 4682  df-rel 4683  df-res 4688

This theorem is referenced by:  elres  4996  resiexg  5005  iss  5006  dfres2  5012  restidsing  5016  issref  5066  asymref  5069  poirr2  5076  cnvcnvres  5147  resco  5188  ressn  5224  funssres  5314  fnresdisj  5387  fnres  5394  fcnvres  5461  nfunsn  5613  fsnunfv  5787  resfunexgALT  6195  setsresg  12903