Theorem relin1 4455

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 3157	. 2 |- ( A i^i B ) C_ A
2		relss 4427	. 2 |- ( ( A i^i B ) C_ A -> ( Rel A -> Rel ( A i^i B ) ) )
3	1, 2	ax-mp 7	1 |- ( Rel A -> Rel ( A i^i B ) )

Syntax hints:   -> wi 4   i^i cin 2916   C_ wss 2917   Rel wrel 4350

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924  df-ss 2931  df-rel 4352

This theorem is referenced by:  inopab  4468