Theorem relin2 4807

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 3398	. 2 |- ( A i^i B ) C_ B
2		relss 4775	. 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 3169   C_ wss 3170   Rel wrel 4693

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-in 3176  df-ss 3183  df-rel 4695

This theorem is referenced by:  intasym  5081  asymref  5082  poirr2  5089