Theorem elpred

Description: Membership in a predecessor class. (Contributed by Scott Fenton, 4-Feb-2011)

	Ref	Expression
Hypothesis	elpred.1	$$⊢Y\in \mathrm{V}$$
Assertion	elpred	$$⊢X\in D\to \left(Y\in Pred\left(R,A,X\right)\leftrightarrow \left(Y\in A\wedge YRX\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		elpred.1	$$⊢Y\in \mathrm{V}$$
2		df-pred	$$⊢Pred\left(R,A,X\right)=A\cap R^{-1}\left[\left\{X\right\}\right]$$
3	2	elin2	$$⊢Y\in Pred\left(R,A,X\right)\leftrightarrow \left(Y\in A\wedge Y\in R^{-1}\left[\left\{X\right\}\right]\right)$$
4	1	eliniseg	$$⊢X\in D\to \left(Y\in R^{-1}\left[\left\{X\right\}\right]\leftrightarrow YRX\right)$$
5	4	anbi2d	$$⊢X\in D\to \left(\left(Y\in A\wedge Y\in R^{-1}\left[\left\{X\right\}\right]\right)\leftrightarrow \left(Y\in A\wedge YRX\right)\right)$$
6	3 5	syl5bb	$$⊢X\in D\to \left(Y\in Pred\left(R,A,X\right)\leftrightarrow \left(Y\in A\wedge YRX\right)\right)$$