Theorem relin1 4665

Description: The intersection with a relation is a relation. (Contributed by NM, 16-Aug-1994.)

Assertion

Ref	Expression
relin1	|- ( Rel A -> Rel ( A i^i B ) )

Proof of Theorem relin1

Step	Hyp	Ref	Expression
1		inss1 3301	. 2 |- ( A i^i B ) C_ A
2		relss 4634	. 2 |- ( ( A i^i B ) C_ A -> ( Rel A -> Rel ( A i^i B ) ) )
3	1, 2	ax-mp 5	1 |- ( Rel A -> Rel ( A i^i B ) )

Syntax hints:   -> wi 4   i^i cin 3075   C_ wss 3076   Rel wrel 4552

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 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-in 3082  df-ss 3089  df-rel 4554

This theorem is referenced by:  inopab  4679