Theorem relres 4891

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 4595	. . 3 |- ( A |` B ) = ( A i^i ( B X. _V ) )
2		inss2 3328	. . 3 |- ( A i^i ( B X. _V ) ) C_ ( B X. _V )
3	1, 2	eqsstri 3160	. 2 |- ( A |` B ) C_ ( B X. _V )
4		relxp 4692	. 2 |- Rel ( B X. _V )
5		relss 4670	. 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 2712   i^i cin 3101   C_ wss 3102   X. cxp 4581   |` cres 4585   Rel wrel 4588

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-in 3108  df-ss 3115  df-opab 4026  df-xp 4589  df-rel 4590  df-res 4595

This theorem is referenced by:  elres  4899  resiexg  4908  iss  4909  dfres2  4915  issref  4965  asymref  4968  poirr2  4975  cnvcnvres  5046  resco  5087  ressn  5123  funssres  5209  fnresdisj  5277  fnres  5283  fcnvres  5350  nfunsn  5499  fsnunfv  5665  resfunexgALT  6052  setsresg  12188