Theorem reldif 5238

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 3737	. 2 |- ( A \ B ) C_ A
2		relss 5206	. 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 3571   C_ wss 3574   Rel wrel 5119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-rel 5121

This theorem is referenced by:  difopab  5253  fundif  5935  relsdom  7962  opeldifid  29412  fundmpss  31664  relbigcup  32004  vvdifopab  34024