Theorem relres 5065

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 4760	. . 3 |- ( A |` B ) = ( A i^i ( B X. _V ) )
2		inss2 3441	. . 3 |- ( A i^i ( B X. _V ) ) C_ ( B X. _V )
3	1, 2	eqsstri 3269	. 2 |- ( A |` B ) C_ ( B X. _V )
4		relxp 4858	. 2 |- Rel ( B X. _V )
5		relss 4836	. 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 2812   i^i cin 3209   C_ wss 3210   X. cxp 4746   |` cres 4750   Rel wrel 4753

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 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2814  df-in 3216  df-ss 3223  df-opab 4171  df-xp 4754  df-rel 4755  df-res 4760

This theorem is referenced by:  elres  5073  resiexg  5082  iss  5083  dfres2  5089  restidsing  5093  issref  5144  asymref  5147  poirr2  5154  cnvcnvres  5225  resco  5266  ressn  5302  funssres  5394  fnresdisj  5467  fnres  5474  fcnvres  5549  nfunsn  5706  fsnunfv  5884  resfunexgALT  6300  setsresg  13242