Theorem relres 5089

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 4784	. . 3 |- ( A |` B ) = ( A i^i ( B X. _V ) )
2		inss2 3452	. . 3 |- ( A i^i ( B X. _V ) ) C_ ( B X. _V )
3	1, 2	eqsstri 3280	. 2 |- ( A |` B ) C_ ( B X. _V )
4		relxp 4882	. 2 |- Rel ( B X. _V )
5		relss 4860	. 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 2821   i^i cin 3219   C_ wss 3220   X. cxp 4770   |` cres 4774   Rel wrel 4777

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226  df-ss 3233  df-opab 4191  df-xp 4778  df-rel 4779  df-res 4784

This theorem is referenced by:  elres  5097  resiexg  5106  iss  5107  dfres2  5113  restidsing  5117  issref  5168  asymref  5171  poirr2  5178  cnvcnvres  5249  resco  5290  ressn  5326  funssres  5418  fnresdisj  5491  fnres  5498  fcnvres  5573  nfunsn  5730  fsnunfv  5910  resfunexgALT  6331  setsresg  13373