Theorem predrelss

Description: Subset carries from relation to predecessor class. (Contributed by Scott Fenton, 25-Nov-2024)

	Ref	Expression
Assertion	predrelss	$$⊢R\subseteq S\to Pred\left(R,A,X\right)\subseteq Pred\left(S,A,X\right)$$

Proof

Step	Hyp	Ref	Expression
1		cnvss	$$⊢R\subseteq S\to R^{-1}\subseteq S^{-1}$$
2		imass1	$$⊢R^{-1}\subseteq S^{-1}\to R^{-1}\left[\left\{X\right\}\right]\subseteq S^{-1}\left[\left\{X\right\}\right]$$
3		sslin	$$⊢R^{-1}\left[\left\{X\right\}\right]\subseteq S^{-1}\left[\left\{X\right\}\right]\to A\cap R^{-1}\left[\left\{X\right\}\right]\subseteq A\cap S^{-1}\left[\left\{X\right\}\right]$$
4	1 2 3	3syl	$$⊢R\subseteq S\to A\cap R^{-1}\left[\left\{X\right\}\right]\subseteq A\cap S^{-1}\left[\left\{X\right\}\right]$$
5		df-pred	$$⊢Pred\left(R,A,X\right)=A\cap R^{-1}\left[\left\{X\right\}\right]$$
6		df-pred	$$⊢Pred\left(S,A,X\right)=A\cap S^{-1}\left[\left\{X\right\}\right]$$
7	4 5 6	3sstr4g	$$⊢R\subseteq S\to Pred\left(R,A,X\right)\subseteq Pred\left(S,A,X\right)$$