Metamath Proof Explorer


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 𝑖𝑛 → ( 𝑓 ‘ suc 𝑖 ) = 𝑦 ∈ ( 𝑓𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
Assertion bnj589 ( 𝜓 ↔ ∀ 𝑘 ∈ ω ( suc 𝑘𝑛 → ( 𝑓 ‘ suc 𝑘 ) = 𝑦 ∈ ( 𝑓𝑘 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )

Proof

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