Theorem bnj1020

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	bnj1020.1	⊢ ( 𝜑 ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) )
	bnj1020.2	⊢ ( 𝜓 ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
	bnj1020.3	⊢ ( 𝜒 ↔ ( 𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) )
	bnj1020.4	⊢ ( 𝜃 ↔ ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑦 ∈ trCl ( 𝑋 , 𝐴 , 𝑅 ) ∧ 𝑧 ∈ pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
	bnj1020.5	⊢ ( 𝜏 ↔ ( 𝑚 ∈ ω ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) )
	bnj1020.6	⊢ ( 𝜂 ↔ ( 𝑖 ∈ 𝑛 ∧ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) ) )
	bnj1020.7	⊢ ( 𝜑′ ↔ [ 𝑝 / 𝑛 ] 𝜑 )
	bnj1020.8	⊢ ( 𝜓′ ↔ [ 𝑝 / 𝑛 ] 𝜓 )
	bnj1020.9	⊢ ( 𝜒′ ↔ [ 𝑝 / 𝑛 ] 𝜒 )
	bnj1020.10	⊢ ( 𝜑″ ↔ [ 𝐺 / 𝑓 ] 𝜑′ )
	bnj1020.11	⊢ ( 𝜓″ ↔ [ 𝐺 / 𝑓 ] 𝜓′ )
	bnj1020.12	⊢ ( 𝜒″ ↔ [ 𝐺 / 𝑓 ] 𝜒′ )
	bnj1020.13	⊢ 𝐷 = ( ω ∖ { ∅ } )
	bnj1020.14	⊢ 𝐵 = { 𝑓 ∣ ∃ 𝑛 ∈ 𝐷 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) }
	bnj1020.15	⊢ 𝐶 = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑚 ) pred ( 𝑦 , 𝐴 , 𝑅 )
	bnj1020.16	⊢ 𝐺 = ( 𝑓 ∪ { ⟨ 𝑛 , 𝐶 ⟩ } )
	bnj1020.26	⊢ ( 𝜒″ ↔ ( 𝑝 ∈ 𝐷 ∧ 𝐺 Fn 𝑝 ∧ 𝜑″ ∧ 𝜓″ ) )
Assertion	bnj1020	⊢ ( ( 𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃ 𝑝 𝜏 ) → pred ( 𝑦 , 𝐴 , 𝑅 ) ⊆ trCl ( 𝑋 , 𝐴 , 𝑅 ) )

Proof

Step	Hyp	Ref	Expression
1		bnj1020.1	⊢ ( 𝜑 ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) )
2		bnj1020.2	⊢ ( 𝜓 ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
3		bnj1020.3	⊢ ( 𝜒 ↔ ( 𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) )
4		bnj1020.4	⊢ ( 𝜃 ↔ ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑦 ∈ trCl ( 𝑋 , 𝐴 , 𝑅 ) ∧ 𝑧 ∈ pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
5		bnj1020.5	⊢ ( 𝜏 ↔ ( 𝑚 ∈ ω ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) )
6		bnj1020.6	⊢ ( 𝜂 ↔ ( 𝑖 ∈ 𝑛 ∧ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) ) )
7		bnj1020.7	⊢ ( 𝜑′ ↔ [ 𝑝 / 𝑛 ] 𝜑 )
8		bnj1020.8	⊢ ( 𝜓′ ↔ [ 𝑝 / 𝑛 ] 𝜓 )
9		bnj1020.9	⊢ ( 𝜒′ ↔ [ 𝑝 / 𝑛 ] 𝜒 )
10		bnj1020.10	⊢ ( 𝜑″ ↔ [ 𝐺 / 𝑓 ] 𝜑′ )
11		bnj1020.11	⊢ ( 𝜓″ ↔ [ 𝐺 / 𝑓 ] 𝜓′ )
12		bnj1020.12	⊢ ( 𝜒″ ↔ [ 𝐺 / 𝑓 ] 𝜒′ )
13		bnj1020.13	⊢ 𝐷 = ( ω ∖ { ∅ } )
14		bnj1020.14	⊢ 𝐵 = { 𝑓 ∣ ∃ 𝑛 ∈ 𝐷 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) }
15		bnj1020.15	⊢ 𝐶 = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑚 ) pred ( 𝑦 , 𝐴 , 𝑅 )
16		bnj1020.16	⊢ 𝐺 = ( 𝑓 ∪ { ⟨ 𝑛 , 𝐶 ⟩ } )
17		bnj1020.26	⊢ ( 𝜒″ ↔ ( 𝑝 ∈ 𝐷 ∧ 𝐺 Fn 𝑝 ∧ 𝜑″ ∧ 𝜓″ ) )
18		bnj1019	⊢ ( ∃ 𝑝 ( 𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂 ) ↔ ( 𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃ 𝑝 𝜏 ) )
19	1 2 3 4 5 7 8 9 10 11 12 13 14 15 16	bnj998	⊢ ( ( 𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂 ) → 𝜒″ )
20	3 5 6 13 19	bnj1001	⊢ ( ( 𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂 ) → ( 𝜒″ ∧ 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ) )
21	1 2 3 4 5 6 7 8 9 10 11 12 13 15 16 20	bnj1006	⊢ ( ( 𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂 ) → pred ( 𝑦 , 𝐴 , 𝑅 ) ⊆ ( 𝐺 ‘ suc 𝑖 ) )
22	21	exlimiv	⊢ ( ∃ 𝑝 ( 𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂 ) → pred ( 𝑦 , 𝐴 , 𝑅 ) ⊆ ( 𝐺 ‘ suc 𝑖 ) )
23	18 22	sylbir	⊢ ( ( 𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃ 𝑝 𝜏 ) → pred ( 𝑦 , 𝐴 , 𝑅 ) ⊆ ( 𝐺 ‘ suc 𝑖 ) )
24	1 2 3 4 5 7 8 9 10 11 12 13 14 15 16 17 19 20	bnj1018	⊢ ( ( 𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃ 𝑝 𝜏 ) → ( 𝐺 ‘ suc 𝑖 ) ⊆ trCl ( 𝑋 , 𝐴 , 𝑅 ) )
25	23 24	sstrd	⊢ ( ( 𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃ 𝑝 𝜏 ) → pred ( 𝑦 , 𝐴 , 𝑅 ) ⊆ trCl ( 𝑋 , 𝐴 , 𝑅 ) )