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	$$⊢A\subseteq B\to Pred\left(R,A,X\right)\subseteq Pred\left(R,B,X\right)$$

Proof

Step	Hyp	Ref	Expression
1		ssrin	$$⊢A\subseteq B\to A\cap R^{-1}\left[\left\{X\right\}\right]\subseteq B\cap R^{-1}\left[\left\{X\right\}\right]$$
2		df-pred	$$⊢Pred\left(R,A,X\right)=A\cap R^{-1}\left[\left\{X\right\}\right]$$
3		df-pred	$$⊢Pred\left(R,B,X\right)=B\cap R^{-1}\left[\left\{X\right\}\right]$$
4	1 2 3	3sstr4g	$$⊢A\subseteq B\to Pred\left(R,A,X\right)\subseteq Pred\left(R,B,X\right)$$