Theorem relin2 4698

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 3324	. 2 |- ( A i^i B ) C_ B
2		relss 4666	. 2 |- ( ( A i^i B ) C_ B -> ( Rel B -> Rel ( A i^i B ) ) )
3	1, 2	ax-mp 5	1 |- ( Rel B -> Rel ( A i^i B ) )

Syntax hints:   -> wi 4   i^i cin 3097   C_ wss 3098   Rel wrel 4584

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-v 2711  df-in 3104  df-ss 3111  df-rel 4586

This theorem is referenced by:  intasym  4963  asymref  4964  poirr2  4971