Theorem relres 5071

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 4766	. . 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 4864	. 2 |- Rel ( B X. _V )
5		relss 4842	. 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 4752   |` cres 4756   Rel wrel 4759

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 4177  df-xp 4760  df-rel 4761  df-res 4766

This theorem is referenced by:  elres  5079  resiexg  5088  iss  5089  dfres2  5095  restidsing  5099  issref  5150  asymref  5153  poirr2  5160  cnvcnvres  5231  resco  5272  ressn  5308  funssres  5400  fnresdisj  5473  fnres  5480  fcnvres  5555  nfunsn  5712  fsnunfv  5890  resfunexgALT  6310  setsresg  13334