Theorem bnj589

Description: Technical lemma for bnj852 . This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

	Ref	Expression
Hypothesis	bnj589.1	$$⊢\psi \leftrightarrow \forall i\in \mathrm{\omega }\left(\mathrm{suc}i\in n\to f\left(\mathrm{suc}i\right)=\bigcup _{y\in f\left(i\right)}pred\left(y,A,R\right)\right)$$
Assertion	bnj589	$$⊢\psi \leftrightarrow \forall k\in \mathrm{\omega }\left(\mathrm{suc}k\in n\to f\left(\mathrm{suc}k\right)=\bigcup _{y\in f\left(k\right)}pred\left(y,A,R\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		bnj589.1	$$⊢\psi \leftrightarrow \forall i\in \mathrm{\omega }\left(\mathrm{suc}i\in n\to f\left(\mathrm{suc}i\right)=\bigcup _{y\in f\left(i\right)}pred\left(y,A,R\right)\right)$$
2	1	bnj222	$$⊢\psi \leftrightarrow \forall k\in \mathrm{\omega }\left(\mathrm{suc}k\in n\to f\left(\mathrm{suc}k\right)=\bigcup _{y\in f\left(k\right)}pred\left(y,A,R\right)\right)$$