Theorem relres 4935

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 4638	. . 3 |- ( A |` B ) = ( A i^i ( B X. _V ) )
2		inss2 3356	. . 3 |- ( A i^i ( B X. _V ) ) C_ ( B X. _V )
3	1, 2	eqsstri 3187	. 2 |- ( A |` B ) C_ ( B X. _V )
4		relxp 4735	. 2 |- Rel ( B X. _V )
5		relss 4713	. 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 2737   i^i cin 3128   C_ wss 3129   X. cxp 4624   |` cres 4628   Rel wrel 4631

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 2739  df-in 3135  df-ss 3142  df-opab 4065  df-xp 4632  df-rel 4633  df-res 4638

This theorem is referenced by:  elres  4943  resiexg  4952  iss  4953  dfres2  4959  restidsing  4963  issref  5011  asymref  5014  poirr2  5021  cnvcnvres  5092  resco  5133  ressn  5169  funssres  5258  fnresdisj  5326  fnres  5332  fcnvres  5399  nfunsn  5549  fsnunfv  5717  resfunexgALT  6108  setsresg  12494