Theorem relres 5072

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 4767	. . 3 |- ( A |` B ) = ( A i^i ( B X. _V ) )
2		inss2 3446	. . 3 |- ( A i^i ( B X. _V ) ) C_ ( B X. _V )
3	1, 2	eqsstri 3274	. 2 |- ( A |` B ) C_ ( B X. _V )
4		relxp 4865	. 2 |- Rel ( B X. _V )
5		relss 4843	. 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 2815   i^i cin 3213   C_ wss 3214   X. cxp 4753   |` cres 4757   Rel wrel 4760

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 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-in 3220  df-ss 3227  df-opab 4178  df-xp 4761  df-rel 4762  df-res 4767

This theorem is referenced by:  elres  5080  resiexg  5089  iss  5090  dfres2  5096  restidsing  5100  issref  5151  asymref  5154  poirr2  5161  cnvcnvres  5232  resco  5273  ressn  5309  funssres  5401  fnresdisj  5474  fnres  5481  fcnvres  5556  nfunsn  5713  fsnunfv  5891  resfunexgALT  6311  setsresg  13339