Theorem relres 5047

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 4743	. . 3 |- ( A |` B ) = ( A i^i ( B X. _V ) )
2		inss2 3430	. . 3 |- ( A i^i ( B X. _V ) ) C_ ( B X. _V )
3	1, 2	eqsstri 3260	. 2 |- ( A |` B ) C_ ( B X. _V )
4		relxp 4841	. 2 |- Rel ( B X. _V )
5		relss 4819	. 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 2803   i^i cin 3200   C_ wss 3201   X. cxp 4729   |` cres 4733   Rel wrel 4736

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-in 3207  df-ss 3214  df-opab 4156  df-xp 4737  df-rel 4738  df-res 4743

This theorem is referenced by:  elres  5055  resiexg  5064  iss  5065  dfres2  5071  restidsing  5075  issref  5126  asymref  5129  poirr2  5136  cnvcnvres  5207  resco  5248  ressn  5284  funssres  5376  fnresdisj  5449  fnres  5456  fcnvres  5528  nfunsn  5685  fsnunfv  5863  resfunexgALT  6279  setsresg  13181