Theorem bnj222

Description: Technical lemma for bnj229 . 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	bnj222.1	⊢ ( 𝜓 ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑁 → ( 𝐹 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝐹 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
Assertion	bnj222	⊢ ( 𝜓 ↔ ∀ 𝑚 ∈ ω ( suc 𝑚 ∈ 𝑁 → ( 𝐹 ‘ suc 𝑚 ) = ∪ 𝑦 ∈ ( 𝐹 ‘ 𝑚 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )

Proof

Step	Hyp	Ref	Expression
1		bnj222.1	⊢ ( 𝜓 ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑁 → ( 𝐹 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝐹 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
2		suceq	⊢ ( 𝑖 = 𝑚 → suc 𝑖 = suc 𝑚 )
3	2	eleq1d	⊢ ( 𝑖 = 𝑚 → ( suc 𝑖 ∈ 𝑁 ↔ suc 𝑚 ∈ 𝑁 ) )
4	2	fveq2d	⊢ ( 𝑖 = 𝑚 → ( 𝐹 ‘ suc 𝑖 ) = ( 𝐹 ‘ suc 𝑚 ) )
5		fveq2	⊢ ( 𝑖 = 𝑚 → ( 𝐹 ‘ 𝑖 ) = ( 𝐹 ‘ 𝑚 ) )
6	5	bnj1113	⊢ ( 𝑖 = 𝑚 → ∪ 𝑦 ∈ ( 𝐹 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) = ∪ 𝑦 ∈ ( 𝐹 ‘ 𝑚 ) pred ( 𝑦 , 𝐴 , 𝑅 ) )
7	4 6	eqeq12d	⊢ ( 𝑖 = 𝑚 → ( ( 𝐹 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝐹 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ↔ ( 𝐹 ‘ suc 𝑚 ) = ∪ 𝑦 ∈ ( 𝐹 ‘ 𝑚 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
8	3 7	imbi12d	⊢ ( 𝑖 = 𝑚 → ( ( suc 𝑖 ∈ 𝑁 → ( 𝐹 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝐹 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ↔ ( suc 𝑚 ∈ 𝑁 → ( 𝐹 ‘ suc 𝑚 ) = ∪ 𝑦 ∈ ( 𝐹 ‘ 𝑚 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) )
9	8	cbvralvw	⊢ ( ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑁 → ( 𝐹 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝐹 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ↔ ∀ 𝑚 ∈ ω ( suc 𝑚 ∈ 𝑁 → ( 𝐹 ‘ suc 𝑚 ) = ∪ 𝑦 ∈ ( 𝐹 ‘ 𝑚 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
10	1 9	bitri	⊢ ( 𝜓 ↔ ∀ 𝑚 ∈ ω ( suc 𝑚 ∈ 𝑁 → ( 𝐹 ‘ suc 𝑚 ) = ∪ 𝑦 ∈ ( 𝐹 ‘ 𝑚 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )