Theorem bnj917

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	bnj917.1	⊢ ( 𝜑 ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) )
	bnj917.2	⊢ ( 𝜓 ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
	bnj917.3	⊢ 𝐷 = ( ω ∖ { ∅ } )
	bnj917.4	⊢ 𝐵 = { 𝑓 ∣ ∃ 𝑛 ∈ 𝐷 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) }
	bnj917.5	⊢ ( 𝜒 ↔ ( 𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) )
Assertion	bnj917	⊢ ( 𝑦 ∈ trCl ( 𝑋 , 𝐴 , 𝑅 ) → ∃ 𝑓 ∃ 𝑛 ∃ 𝑖 ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) ) )

Proof

Step	Hyp	Ref	Expression
1		bnj917.1	⊢ ( 𝜑 ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) )
2		bnj917.2	⊢ ( 𝜓 ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
3		bnj917.3	⊢ 𝐷 = ( ω ∖ { ∅ } )
4		bnj917.4	⊢ 𝐵 = { 𝑓 ∣ ∃ 𝑛 ∈ 𝐷 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) }
5		bnj917.5	⊢ ( 𝜒 ↔ ( 𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) )
6		biid	⊢ ( ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ↔ ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) )
7	1 2 3 4 6	bnj916	⊢ ( 𝑦 ∈ trCl ( 𝑋 , 𝐴 , 𝑅 ) → ∃ 𝑓 ∃ 𝑛 ∃ 𝑖 ( 𝑛 ∈ 𝐷 ∧ ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ∧ 𝑖 ∈ 𝑛 ∧ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) ) )
8		bnj252	⊢ ( ( 𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ↔ ( 𝑛 ∈ 𝐷 ∧ ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ) )
9	5 8	bitri	⊢ ( 𝜒 ↔ ( 𝑛 ∈ 𝐷 ∧ ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ) )
10	9	3anbi1i	⊢ ( ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) ) ↔ ( ( 𝑛 ∈ 𝐷 ∧ ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ) ∧ 𝑖 ∈ 𝑛 ∧ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) ) )
11		bnj253	⊢ ( ( 𝑛 ∈ 𝐷 ∧ ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ∧ 𝑖 ∈ 𝑛 ∧ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) ) ↔ ( ( 𝑛 ∈ 𝐷 ∧ ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ) ∧ 𝑖 ∈ 𝑛 ∧ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) ) )
12	10 11	bitr4i	⊢ ( ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) ) ↔ ( 𝑛 ∈ 𝐷 ∧ ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ∧ 𝑖 ∈ 𝑛 ∧ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) ) )
13	12	3exbii	⊢ ( ∃ 𝑓 ∃ 𝑛 ∃ 𝑖 ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) ) ↔ ∃ 𝑓 ∃ 𝑛 ∃ 𝑖 ( 𝑛 ∈ 𝐷 ∧ ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ∧ 𝑖 ∈ 𝑛 ∧ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) ) )
14	7 13	sylibr	⊢ ( 𝑦 ∈ trCl ( 𝑋 , 𝐴 , 𝑅 ) → ∃ 𝑓 ∃ 𝑛 ∃ 𝑖 ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) ) )