Theorem relin2 4474

Description: The intersection with a relation is a relation. (Contributed by NM, 17-Jan-2006.)

Assertion

Ref	Expression
relin2	|- ( Rel B -> Rel ( A i^i B ) )

Proof of Theorem relin2

Step	Hyp	Ref	Expression
1		inss2 3187	. 2 |- ( A i^i B ) C_ B
2		relss 4445	. 2 |- ( ( A i^i B ) C_ B -> ( Rel B -> Rel ( A i^i B ) ) )
3	1, 2	ax-mp 7	1 |- ( Rel B -> Rel ( A i^i B ) )

Syntax hints:   -> wi 4   i^i cin 2972   C_ wss 2973   Rel wrel 4368

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-in 2979  df-ss 2986  df-rel 4370

This theorem is referenced by:  intasym  4729  asymref  4730  poirr2  4737