Theorem predidm

Description: Idempotent law for the predecessor class. (Contributed by Scott Fenton, 29-Mar-2011)

	Ref	Expression
Assertion	predidm	$$⊢Pred\left(R,Pred\left(R,A,X\right),X\right)=Pred\left(R,A,X\right)$$

Proof

Step	Hyp	Ref	Expression
1		df-pred	$$⊢Pred\left(R,Pred\left(R,A,X\right),X\right)=Pred\left(R,A,X\right)\cap R^{-1}\left[\left\{X\right\}\right]$$
2		df-pred	$$⊢Pred\left(R,A,X\right)=A\cap R^{-1}\left[\left\{X\right\}\right]$$
3		inidm	$$⊢R^{-1}\left[\left\{X\right\}\right]\cap R^{-1}\left[\left\{X\right\}\right]=R^{-1}\left[\left\{X\right\}\right]$$
4	3	ineq2i	$$⊢A\cap \left(R^{-1}\left[\left\{X\right\}\right]\cap R^{-1}\left[\left\{X\right\}\right]\right)=A\cap R^{-1}\left[\left\{X\right\}\right]$$
5	2 4	eqtr4i	$$⊢Pred\left(R,A,X\right)=A\cap \left(R^{-1}\left[\left\{X\right\}\right]\cap R^{-1}\left[\left\{X\right\}\right]\right)$$
6		inass	$$⊢\left(A\cap R^{-1}\left[\left\{X\right\}\right]\right)\cap R^{-1}\left[\left\{X\right\}\right]=A\cap \left(R^{-1}\left[\left\{X\right\}\right]\cap R^{-1}\left[\left\{X\right\}\right]\right)$$
7	5 6	eqtr4i	$$⊢Pred\left(R,A,X\right)=\left(A\cap R^{-1}\left[\left\{X\right\}\right]\right)\cap R^{-1}\left[\left\{X\right\}\right]$$
8	2	ineq1i	$$⊢Pred\left(R,A,X\right)\cap R^{-1}\left[\left\{X\right\}\right]=\left(A\cap R^{-1}\left[\left\{X\right\}\right]\right)\cap R^{-1}\left[\left\{X\right\}\right]$$
9	7 8	eqtr4i	$$⊢Pred\left(R,A,X\right)=Pred\left(R,A,X\right)\cap R^{-1}\left[\left\{X\right\}\right]$$
10	1 9	eqtr4i	$$⊢Pred\left(R,Pred\left(R,A,X\right),X\right)=Pred\left(R,A,X\right)$$