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	⊢ ( 𝜓 ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
Assertion	bnj589	⊢ ( 𝜓 ↔ ∀ 𝑘 ∈ ω ( suc 𝑘 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑘 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑘 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )

Proof

Step	Hyp	Ref	Expression
1		bnj589.1	⊢ ( 𝜓 ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
2	1	bnj222	⊢ ( 𝜓 ↔ ∀ 𝑘 ∈ ω ( suc 𝑘 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑘 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑘 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )