Theorem bnj1112

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
Hypothesis	bnj1112.1	⊢ ( 𝜓 ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
Assertion	bnj1112	⊢ ( 𝜓 ↔ ∀ 𝑗 ( ( 𝑗 ∈ ω ∧ suc 𝑗 ∈ 𝑛 ) → ( 𝑓 ‘ suc 𝑗 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑗 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )

Proof

Step	Hyp	Ref	Expression
1		bnj1112.1	⊢ ( 𝜓 ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
2	1	bnj115	⊢ ( 𝜓 ↔ ∀ 𝑖 ( ( 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛 ) → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
3		eleq1w	⊢ ( 𝑖 = 𝑗 → ( 𝑖 ∈ ω ↔ 𝑗 ∈ ω ) )
4		suceq	⊢ ( 𝑖 = 𝑗 → suc 𝑖 = suc 𝑗 )
5	4	eleq1d	⊢ ( 𝑖 = 𝑗 → ( suc 𝑖 ∈ 𝑛 ↔ suc 𝑗 ∈ 𝑛 ) )
6	3 5	anbi12d	⊢ ( 𝑖 = 𝑗 → ( ( 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛 ) ↔ ( 𝑗 ∈ ω ∧ suc 𝑗 ∈ 𝑛 ) ) )
7	4	fveq2d	⊢ ( 𝑖 = 𝑗 → ( 𝑓 ‘ suc 𝑖 ) = ( 𝑓 ‘ suc 𝑗 ) )
8		fveq2	⊢ ( 𝑖 = 𝑗 → ( 𝑓 ‘ 𝑖 ) = ( 𝑓 ‘ 𝑗 ) )
9	8	bnj1113	⊢ ( 𝑖 = 𝑗 → ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑗 ) pred ( 𝑦 , 𝐴 , 𝑅 ) )
10	7 9	eqeq12d	⊢ ( 𝑖 = 𝑗 → ( ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ↔ ( 𝑓 ‘ suc 𝑗 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑗 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
11	6 10	imbi12d	⊢ ( 𝑖 = 𝑗 → ( ( ( 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛 ) → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ↔ ( ( 𝑗 ∈ ω ∧ suc 𝑗 ∈ 𝑛 ) → ( 𝑓 ‘ suc 𝑗 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑗 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) )
12	11	cbvalvw	⊢ ( ∀ 𝑖 ( ( 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛 ) → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ↔ ∀ 𝑗 ( ( 𝑗 ∈ ω ∧ suc 𝑗 ∈ 𝑛 ) → ( 𝑓 ‘ suc 𝑗 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑗 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
13	2 12	bitri	⊢ ( 𝜓 ↔ ∀ 𝑗 ( ( 𝑗 ∈ ω ∧ suc 𝑗 ∈ 𝑛 ) → ( 𝑓 ‘ suc 𝑗 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑗 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )