Theorem predpredss

Description: If A is a subset of B , then their predecessor classes are also subsets. (Contributed by Scott Fenton, 2-Feb-2011)

	Ref	Expression
Assertion	predpredss	|- ( A C_ B -> Pred ( R , A , X ) C_ Pred ( R , B , X ) )

Proof

Step	Hyp	Ref	Expression
1		ssrin	|- ( A C_ B -> ( A i^i ( `' R " { X } ) ) C_ ( B i^i ( `' R " { X } ) ) )
2		df-pred	|- Pred ( R , A , X ) = ( A i^i ( `' R " { X } ) )
3		df-pred	|- Pred ( R , B , X ) = ( B i^i ( `' R " { X } ) )
4	1 2 3	3sstr4g	|- ( A C_ B -> Pred ( R , A , X ) C_ Pred ( R , B , X ) )