Theorem reldif 4838

Description: A difference cutting down a relation is a relation. (Contributed by NM, 31-Mar-1998.)

Assertion

Ref	Expression
reldif	|- ( Rel A -> Rel ( A \ B ) )

Proof of Theorem reldif

Step	Hyp	Ref	Expression
1		difss 3330	. 2 |- ( A \ B ) C_ A
2		relss 4805	. 2 |- ( ( A \ B ) C_ A -> ( Rel A -> Rel ( A \ B ) ) )
3	1, 2	ax-mp 5	1 |- ( Rel A -> Rel ( A \ B ) )

Syntax hints:   -> wi 4   \ cdif 3194   C_ wss 3197   Rel wrel 4723

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-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-dif 3199  df-in 3203  df-ss 3210  df-rel 4725

This theorem is referenced by:  difopab  4854  fundif  5364