Theorem bnj965

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
Hypotheses	bnj965.1	⊢ ( 𝜓 ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑁 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
	bnj965.2	⊢ ( 𝜓″ ↔ [ 𝐺 / 𝑓 ] 𝜓 )
	bnj965.12000	⊢ 𝐶 = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑚 ) pred ( 𝑦 , 𝐴 , 𝑅 )
	bnj965.13000	⊢ 𝐺 = ( 𝑓 ∪ { ⟨ 𝑛 , 𝐶 ⟩ } )
Assertion	bnj965	⊢ ( 𝜓″ ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑁 → ( 𝐺 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝐺 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )

Proof

Step	Hyp	Ref	Expression
1		bnj965.1	⊢ ( 𝜓 ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑁 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
2		bnj965.2	⊢ ( 𝜓″ ↔ [ 𝐺 / 𝑓 ] 𝜓 )
3		bnj965.12000	⊢ 𝐶 = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑚 ) pred ( 𝑦 , 𝐴 , 𝑅 )
4		bnj965.13000	⊢ 𝐺 = ( 𝑓 ∪ { ⟨ 𝑛 , 𝐶 ⟩ } )
5	4	bnj918	⊢ 𝐺 ∈ V
6	1 2 5 3 4	bnj1000	⊢ ( 𝜓″ ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑁 → ( 𝐺 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝐺 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )