Theorem bnj911

Description: Technical lemma for bnj69 . 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
Hypotheses	bnj911.1	⊢ ( 𝜑 ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) )
	bnj911.2	⊢ ( 𝜓 ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
Assertion	bnj911	⊢ ( ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) → ∀ 𝑖 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		bnj911.1	⊢ ( 𝜑 ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) )
2		bnj911.2	⊢ ( 𝜓 ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
3	2	bnj1095	⊢ ( 𝜓 → ∀ 𝑖 𝜓 )
4	3	bnj1350	⊢ ( ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) → ∀ 𝑖 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) )