Theorem relres 4936

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 4639	. . 3 |- ( A |` B ) = ( A i^i ( B X. _V ) )
2		inss2 3357	. . 3 |- ( A i^i ( B X. _V ) ) C_ ( B X. _V )
3	1, 2	eqsstri 3188	. 2 |- ( A |` B ) C_ ( B X. _V )
4		relxp 4736	. 2 |- Rel ( B X. _V )
5		relss 4714	. 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 2738   i^i cin 3129   C_ wss 3130   X. cxp 4625   |` cres 4629   Rel wrel 4632

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2740  df-in 3136  df-ss 3143  df-opab 4066  df-xp 4633  df-rel 4634  df-res 4639

This theorem is referenced by:  elres  4944  resiexg  4953  iss  4954  dfres2  4960  restidsing  4964  issref  5012  asymref  5015  poirr2  5022  cnvcnvres  5093  resco  5134  ressn  5170  funssres  5259  fnresdisj  5327  fnres  5333  fcnvres  5400  nfunsn  5550  fsnunfv  5718  resfunexgALT  6109  setsresg  12500