Theorem bnj1021

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	bnj1021.1	⊢ ( 𝜑 ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) )
	bnj1021.2	⊢ ( 𝜓 ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
	bnj1021.3	⊢ ( 𝜒 ↔ ( 𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) )
	bnj1021.4	⊢ ( 𝜃 ↔ ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑦 ∈ trCl ( 𝑋 , 𝐴 , 𝑅 ) ∧ 𝑧 ∈ pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
	bnj1021.5	⊢ ( 𝜏 ↔ ( 𝑚 ∈ ω ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) )
	bnj1021.6	⊢ ( 𝜂 ↔ ( 𝑖 ∈ 𝑛 ∧ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) ) )
	bnj1021.13	⊢ 𝐷 = ( ω ∖ { ∅ } )
	bnj1021.14	⊢ 𝐵 = { 𝑓 ∣ ∃ 𝑛 ∈ 𝐷 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) }
Assertion	bnj1021	⊢ ∃ 𝑓 ∃ 𝑛 ∃ 𝑖 ∃ 𝑚 ( 𝜃 → ( 𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃ 𝑝 𝜏 ) )

Proof

Step	Hyp	Ref	Expression
1		bnj1021.1	⊢ ( 𝜑 ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) )
2		bnj1021.2	⊢ ( 𝜓 ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
3		bnj1021.3	⊢ ( 𝜒 ↔ ( 𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) )
4		bnj1021.4	⊢ ( 𝜃 ↔ ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑦 ∈ trCl ( 𝑋 , 𝐴 , 𝑅 ) ∧ 𝑧 ∈ pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
5		bnj1021.5	⊢ ( 𝜏 ↔ ( 𝑚 ∈ ω ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) )
6		bnj1021.6	⊢ ( 𝜂 ↔ ( 𝑖 ∈ 𝑛 ∧ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) ) )
7		bnj1021.13	⊢ 𝐷 = ( ω ∖ { ∅ } )
8		bnj1021.14	⊢ 𝐵 = { 𝑓 ∣ ∃ 𝑛 ∈ 𝐷 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) }
9	1 2 3 4 5 6 7 8	bnj996	⊢ ∃ 𝑓 ∃ 𝑛 ∃ 𝑖 ∃ 𝑚 ∃ 𝑝 ( 𝜃 → ( 𝜒 ∧ 𝜏 ∧ 𝜂 ) )
10		anclb	⊢ ( ( 𝜃 → ( 𝜒 ∧ 𝜏 ∧ 𝜂 ) ) ↔ ( 𝜃 → ( 𝜃 ∧ ( 𝜒 ∧ 𝜏 ∧ 𝜂 ) ) ) )
11		bnj252	⊢ ( ( 𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂 ) ↔ ( 𝜃 ∧ ( 𝜒 ∧ 𝜏 ∧ 𝜂 ) ) )
12	11	imbi2i	⊢ ( ( 𝜃 → ( 𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂 ) ) ↔ ( 𝜃 → ( 𝜃 ∧ ( 𝜒 ∧ 𝜏 ∧ 𝜂 ) ) ) )
13	10 12	bitr4i	⊢ ( ( 𝜃 → ( 𝜒 ∧ 𝜏 ∧ 𝜂 ) ) ↔ ( 𝜃 → ( 𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂 ) ) )
14	13	2exbii	⊢ ( ∃ 𝑚 ∃ 𝑝 ( 𝜃 → ( 𝜒 ∧ 𝜏 ∧ 𝜂 ) ) ↔ ∃ 𝑚 ∃ 𝑝 ( 𝜃 → ( 𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂 ) ) )
15	14	3exbii	⊢ ( ∃ 𝑓 ∃ 𝑛 ∃ 𝑖 ∃ 𝑚 ∃ 𝑝 ( 𝜃 → ( 𝜒 ∧ 𝜏 ∧ 𝜂 ) ) ↔ ∃ 𝑓 ∃ 𝑛 ∃ 𝑖 ∃ 𝑚 ∃ 𝑝 ( 𝜃 → ( 𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂 ) ) )
16	9 15	mpbi	⊢ ∃ 𝑓 ∃ 𝑛 ∃ 𝑖 ∃ 𝑚 ∃ 𝑝 ( 𝜃 → ( 𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂 ) )
17		19.37v	⊢ ( ∃ 𝑝 ( 𝜃 → ( 𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂 ) ) ↔ ( 𝜃 → ∃ 𝑝 ( 𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂 ) ) )
18		bnj1019	⊢ ( ∃ 𝑝 ( 𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂 ) ↔ ( 𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃ 𝑝 𝜏 ) )
19	18	imbi2i	⊢ ( ( 𝜃 → ∃ 𝑝 ( 𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂 ) ) ↔ ( 𝜃 → ( 𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃ 𝑝 𝜏 ) ) )
20	17 19	bitri	⊢ ( ∃ 𝑝 ( 𝜃 → ( 𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂 ) ) ↔ ( 𝜃 → ( 𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃ 𝑝 𝜏 ) ) )
21	20	2exbii	⊢ ( ∃ 𝑖 ∃ 𝑚 ∃ 𝑝 ( 𝜃 → ( 𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂 ) ) ↔ ∃ 𝑖 ∃ 𝑚 ( 𝜃 → ( 𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃ 𝑝 𝜏 ) ) )
22	21	2exbii	⊢ ( ∃ 𝑓 ∃ 𝑛 ∃ 𝑖 ∃ 𝑚 ∃ 𝑝 ( 𝜃 → ( 𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂 ) ) ↔ ∃ 𝑓 ∃ 𝑛 ∃ 𝑖 ∃ 𝑚 ( 𝜃 → ( 𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃ 𝑝 𝜏 ) ) )
23	16 22	mpbi	⊢ ∃ 𝑓 ∃ 𝑛 ∃ 𝑖 ∃ 𝑚 ( 𝜃 → ( 𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃ 𝑝 𝜏 ) )