Theorem bnj590

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	bnj590.1	⊢ ( 𝜓 ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
Assertion	bnj590	⊢ ( ( 𝐵 = suc 𝑖 ∧ 𝜓 ) → ( 𝑖 ∈ ω → ( 𝐵 ∈ 𝑛 → ( 𝑓 ‘ 𝐵 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		bnj590.1	⊢ ( 𝜓 ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
2		rsp	⊢ ( ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) → ( 𝑖 ∈ ω → ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) )
3	1 2	sylbi	⊢ ( 𝜓 → ( 𝑖 ∈ ω → ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) )
4		eleq1	⊢ ( 𝐵 = suc 𝑖 → ( 𝐵 ∈ 𝑛 ↔ suc 𝑖 ∈ 𝑛 ) )
5		fveqeq2	⊢ ( 𝐵 = suc 𝑖 → ( ( 𝑓 ‘ 𝐵 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ↔ ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
6	4 5	imbi12d	⊢ ( 𝐵 = suc 𝑖 → ( ( 𝐵 ∈ 𝑛 → ( 𝑓 ‘ 𝐵 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ↔ ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) )
7	6	imbi2d	⊢ ( 𝐵 = suc 𝑖 → ( ( 𝑖 ∈ ω → ( 𝐵 ∈ 𝑛 → ( 𝑓 ‘ 𝐵 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) ↔ ( 𝑖 ∈ ω → ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) ) )
8	3 7	syl5ibr	⊢ ( 𝐵 = suc 𝑖 → ( 𝜓 → ( 𝑖 ∈ ω → ( 𝐵 ∈ 𝑛 → ( 𝑓 ‘ 𝐵 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) ) )
9	8	imp	⊢ ( ( 𝐵 = suc 𝑖 ∧ 𝜓 ) → ( 𝑖 ∈ ω → ( 𝐵 ∈ 𝑛 → ( 𝑓 ‘ 𝐵 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) )