Theorem bnj998

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	bnj998.1	⊢ ( 𝜑 ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) )
	bnj998.2	⊢ ( 𝜓 ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
	bnj998.3	⊢ ( 𝜒 ↔ ( 𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) )
	bnj998.4	⊢ ( 𝜃 ↔ ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑦 ∈ trCl ( 𝑋 , 𝐴 , 𝑅 ) ∧ 𝑧 ∈ pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
	bnj998.5	⊢ ( 𝜏 ↔ ( 𝑚 ∈ ω ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) )
	bnj998.7	⊢ ( 𝜑′ ↔ [ 𝑝 / 𝑛 ] 𝜑 )
	bnj998.8	⊢ ( 𝜓′ ↔ [ 𝑝 / 𝑛 ] 𝜓 )
	bnj998.9	⊢ ( 𝜒′ ↔ [ 𝑝 / 𝑛 ] 𝜒 )
	bnj998.10	⊢ ( 𝜑″ ↔ [ 𝐺 / 𝑓 ] 𝜑′ )
	bnj998.11	⊢ ( 𝜓″ ↔ [ 𝐺 / 𝑓 ] 𝜓′ )
	bnj998.12	⊢ ( 𝜒″ ↔ [ 𝐺 / 𝑓 ] 𝜒′ )
	bnj998.13	⊢ 𝐷 = ( ω ∖ { ∅ } )
	bnj998.14	⊢ 𝐵 = { 𝑓 ∣ ∃ 𝑛 ∈ 𝐷 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) }
	bnj998.15	⊢ 𝐶 = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑚 ) pred ( 𝑦 , 𝐴 , 𝑅 )
	bnj998.16	⊢ 𝐺 = ( 𝑓 ∪ { ⟨ 𝑛 , 𝐶 ⟩ } )
Assertion	bnj998	⊢ ( ( 𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂 ) → 𝜒″ )

Proof

Step	Hyp	Ref	Expression
1		bnj998.1	⊢ ( 𝜑 ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) )
2		bnj998.2	⊢ ( 𝜓 ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
3		bnj998.3	⊢ ( 𝜒 ↔ ( 𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) )
4		bnj998.4	⊢ ( 𝜃 ↔ ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑦 ∈ trCl ( 𝑋 , 𝐴 , 𝑅 ) ∧ 𝑧 ∈ pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
5		bnj998.5	⊢ ( 𝜏 ↔ ( 𝑚 ∈ ω ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) )
6		bnj998.7	⊢ ( 𝜑′ ↔ [ 𝑝 / 𝑛 ] 𝜑 )
7		bnj998.8	⊢ ( 𝜓′ ↔ [ 𝑝 / 𝑛 ] 𝜓 )
8		bnj998.9	⊢ ( 𝜒′ ↔ [ 𝑝 / 𝑛 ] 𝜒 )
9		bnj998.10	⊢ ( 𝜑″ ↔ [ 𝐺 / 𝑓 ] 𝜑′ )
10		bnj998.11	⊢ ( 𝜓″ ↔ [ 𝐺 / 𝑓 ] 𝜓′ )
11		bnj998.12	⊢ ( 𝜒″ ↔ [ 𝐺 / 𝑓 ] 𝜒′ )
12		bnj998.13	⊢ 𝐷 = ( ω ∖ { ∅ } )
13		bnj998.14	⊢ 𝐵 = { 𝑓 ∣ ∃ 𝑛 ∈ 𝐷 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) }
14		bnj998.15	⊢ 𝐶 = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑚 ) pred ( 𝑦 , 𝐴 , 𝑅 )
15		bnj998.16	⊢ 𝐺 = ( 𝑓 ∪ { ⟨ 𝑛 , 𝐶 ⟩ } )
16		bnj253	⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑦 ∈ trCl ( 𝑋 , 𝐴 , 𝑅 ) ∧ 𝑧 ∈ pred ( 𝑦 , 𝐴 , 𝑅 ) ) ↔ ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ 𝑦 ∈ trCl ( 𝑋 , 𝐴 , 𝑅 ) ∧ 𝑧 ∈ pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
17	16	simp1bi	⊢ ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑦 ∈ trCl ( 𝑋 , 𝐴 , 𝑅 ) ∧ 𝑧 ∈ pred ( 𝑦 , 𝐴 , 𝑅 ) ) → ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) )
18	4 17	sylbi	⊢ ( 𝜃 → ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) )
19	18	bnj705	⊢ ( ( 𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂 ) → ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) )
20		bnj643	⊢ ( ( 𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂 ) → 𝜒 )
21		3simpc	⊢ ( ( 𝑚 ∈ ω ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) → ( 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) )
22	5 21	sylbi	⊢ ( 𝜏 → ( 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) )
23	22	bnj707	⊢ ( ( 𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂 ) → ( 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) )
24		bnj255	⊢ ( ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) ↔ ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ 𝜒 ∧ ( 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) ) )
25	19 20 23 24	syl3anbrc	⊢ ( ( 𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂 ) → ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) )
26		bnj252	⊢ ( ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) ↔ ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) ) )
27	25 26	sylib	⊢ ( ( 𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂 ) → ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) ) )
28		biid	⊢ ( ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ↔ ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) )
29		biid	⊢ ( ( 𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑚 ∈ 𝑛 ) ↔ ( 𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑚 ∈ 𝑛 ) )
30	1 2 3 6 7 8 9 10 11 12 13 14 15 28 29	bnj910	⊢ ( ( ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ∧ ( 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) ) → 𝜒″ )
31	27 30	syl	⊢ ( ( 𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂 ) → 𝜒″ )