Theorem bnj996

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	bnj996.1	⊢ ( 𝜑 ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) )
	bnj996.2	⊢ ( 𝜓 ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
	bnj996.3	⊢ ( 𝜒 ↔ ( 𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) )
	bnj996.4	⊢ ( 𝜃 ↔ ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑦 ∈ trCl ( 𝑋 , 𝐴 , 𝑅 ) ∧ 𝑧 ∈ pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
	bnj996.5	⊢ ( 𝜏 ↔ ( 𝑚 ∈ ω ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) )
	bnj996.6	⊢ ( 𝜂 ↔ ( 𝑖 ∈ 𝑛 ∧ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) ) )
	bnj996.13	⊢ 𝐷 = ( ω ∖ { ∅ } )
	bnj996.14	⊢ 𝐵 = { 𝑓 ∣ ∃ 𝑛 ∈ 𝐷 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) }
Assertion	bnj996	⊢ ∃ 𝑓 ∃ 𝑛 ∃ 𝑖 ∃ 𝑚 ∃ 𝑝 ( 𝜃 → ( 𝜒 ∧ 𝜏 ∧ 𝜂 ) )

Proof

Step	Hyp	Ref	Expression
1		bnj996.1	⊢ ( 𝜑 ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) )
2		bnj996.2	⊢ ( 𝜓 ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
3		bnj996.3	⊢ ( 𝜒 ↔ ( 𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) )
4		bnj996.4	⊢ ( 𝜃 ↔ ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑦 ∈ trCl ( 𝑋 , 𝐴 , 𝑅 ) ∧ 𝑧 ∈ pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
5		bnj996.5	⊢ ( 𝜏 ↔ ( 𝑚 ∈ ω ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛 ) )
6		bnj996.6	⊢ ( 𝜂 ↔ ( 𝑖 ∈ 𝑛 ∧ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) ) )
7		bnj996.13	⊢ 𝐷 = ( ω ∖ { ∅ } )
8		bnj996.14	⊢ 𝐵 = { 𝑓 ∣ ∃ 𝑛 ∈ 𝐷 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) }
9	1 2 7 8 3	bnj917	⊢ ( 𝑦 ∈ trCl ( 𝑋 , 𝐴 , 𝑅 ) → ∃ 𝑓 ∃ 𝑛 ∃ 𝑖 ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) ) )
10	4 9	bnj771	⊢ ( 𝜃 → ∃ 𝑓 ∃ 𝑛 ∃ 𝑖 ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) ) )
11		3anass	⊢ ( ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) ) ↔ ( 𝜒 ∧ ( 𝑖 ∈ 𝑛 ∧ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) ) ) )
12	6	anbi2i	⊢ ( ( 𝜒 ∧ 𝜂 ) ↔ ( 𝜒 ∧ ( 𝑖 ∈ 𝑛 ∧ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) ) ) )
13	11 12	bitr4i	⊢ ( ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) ) ↔ ( 𝜒 ∧ 𝜂 ) )
14	13	3exbii	⊢ ( ∃ 𝑓 ∃ 𝑛 ∃ 𝑖 ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) ) ↔ ∃ 𝑓 ∃ 𝑛 ∃ 𝑖 ( 𝜒 ∧ 𝜂 ) )
15	10 14	sylib	⊢ ( 𝜃 → ∃ 𝑓 ∃ 𝑛 ∃ 𝑖 ( 𝜒 ∧ 𝜂 ) )
16	3 7 5	bnj986	⊢ ( 𝜒 → ∃ 𝑚 ∃ 𝑝 𝜏 )
17	16	ancli	⊢ ( 𝜒 → ( 𝜒 ∧ ∃ 𝑚 ∃ 𝑝 𝜏 ) )
18		19.42vv	⊢ ( ∃ 𝑚 ∃ 𝑝 ( 𝜒 ∧ 𝜏 ) ↔ ( 𝜒 ∧ ∃ 𝑚 ∃ 𝑝 𝜏 ) )
19	17 18	sylibr	⊢ ( 𝜒 → ∃ 𝑚 ∃ 𝑝 ( 𝜒 ∧ 𝜏 ) )
20	19	anim1i	⊢ ( ( 𝜒 ∧ 𝜂 ) → ( ∃ 𝑚 ∃ 𝑝 ( 𝜒 ∧ 𝜏 ) ∧ 𝜂 ) )
21		19.41vv	⊢ ( ∃ 𝑚 ∃ 𝑝 ( ( 𝜒 ∧ 𝜏 ) ∧ 𝜂 ) ↔ ( ∃ 𝑚 ∃ 𝑝 ( 𝜒 ∧ 𝜏 ) ∧ 𝜂 ) )
22	20 21	sylibr	⊢ ( ( 𝜒 ∧ 𝜂 ) → ∃ 𝑚 ∃ 𝑝 ( ( 𝜒 ∧ 𝜏 ) ∧ 𝜂 ) )
23		df-3an	⊢ ( ( 𝜒 ∧ 𝜏 ∧ 𝜂 ) ↔ ( ( 𝜒 ∧ 𝜏 ) ∧ 𝜂 ) )
24	23	2exbii	⊢ ( ∃ 𝑚 ∃ 𝑝 ( 𝜒 ∧ 𝜏 ∧ 𝜂 ) ↔ ∃ 𝑚 ∃ 𝑝 ( ( 𝜒 ∧ 𝜏 ) ∧ 𝜂 ) )
25	22 24	sylibr	⊢ ( ( 𝜒 ∧ 𝜂 ) → ∃ 𝑚 ∃ 𝑝 ( 𝜒 ∧ 𝜏 ∧ 𝜂 ) )
26	25	2eximi	⊢ ( ∃ 𝑛 ∃ 𝑖 ( 𝜒 ∧ 𝜂 ) → ∃ 𝑛 ∃ 𝑖 ∃ 𝑚 ∃ 𝑝 ( 𝜒 ∧ 𝜏 ∧ 𝜂 ) )
27	15 26	bnj593	⊢ ( 𝜃 → ∃ 𝑓 ∃ 𝑛 ∃ 𝑖 ∃ 𝑚 ∃ 𝑝 ( 𝜒 ∧ 𝜏 ∧ 𝜂 ) )
28		19.37v	⊢ ( ∃ 𝑝 ( 𝜃 → ( 𝜒 ∧ 𝜏 ∧ 𝜂 ) ) ↔ ( 𝜃 → ∃ 𝑝 ( 𝜒 ∧ 𝜏 ∧ 𝜂 ) ) )
29	28	exbii	⊢ ( ∃ 𝑚 ∃ 𝑝 ( 𝜃 → ( 𝜒 ∧ 𝜏 ∧ 𝜂 ) ) ↔ ∃ 𝑚 ( 𝜃 → ∃ 𝑝 ( 𝜒 ∧ 𝜏 ∧ 𝜂 ) ) )
30	29	bnj132	⊢ ( ∃ 𝑚 ∃ 𝑝 ( 𝜃 → ( 𝜒 ∧ 𝜏 ∧ 𝜂 ) ) ↔ ( 𝜃 → ∃ 𝑚 ∃ 𝑝 ( 𝜒 ∧ 𝜏 ∧ 𝜂 ) ) )
31	30	exbii	⊢ ( ∃ 𝑖 ∃ 𝑚 ∃ 𝑝 ( 𝜃 → ( 𝜒 ∧ 𝜏 ∧ 𝜂 ) ) ↔ ∃ 𝑖 ( 𝜃 → ∃ 𝑚 ∃ 𝑝 ( 𝜒 ∧ 𝜏 ∧ 𝜂 ) ) )
32	31	bnj132	⊢ ( ∃ 𝑖 ∃ 𝑚 ∃ 𝑝 ( 𝜃 → ( 𝜒 ∧ 𝜏 ∧ 𝜂 ) ) ↔ ( 𝜃 → ∃ 𝑖 ∃ 𝑚 ∃ 𝑝 ( 𝜒 ∧ 𝜏 ∧ 𝜂 ) ) )
33	32	exbii	⊢ ( ∃ 𝑛 ∃ 𝑖 ∃ 𝑚 ∃ 𝑝 ( 𝜃 → ( 𝜒 ∧ 𝜏 ∧ 𝜂 ) ) ↔ ∃ 𝑛 ( 𝜃 → ∃ 𝑖 ∃ 𝑚 ∃ 𝑝 ( 𝜒 ∧ 𝜏 ∧ 𝜂 ) ) )
34	33	bnj132	⊢ ( ∃ 𝑛 ∃ 𝑖 ∃ 𝑚 ∃ 𝑝 ( 𝜃 → ( 𝜒 ∧ 𝜏 ∧ 𝜂 ) ) ↔ ( 𝜃 → ∃ 𝑛 ∃ 𝑖 ∃ 𝑚 ∃ 𝑝 ( 𝜒 ∧ 𝜏 ∧ 𝜂 ) ) )
35	34	exbii	⊢ ( ∃ 𝑓 ∃ 𝑛 ∃ 𝑖 ∃ 𝑚 ∃ 𝑝 ( 𝜃 → ( 𝜒 ∧ 𝜏 ∧ 𝜂 ) ) ↔ ∃ 𝑓 ( 𝜃 → ∃ 𝑛 ∃ 𝑖 ∃ 𝑚 ∃ 𝑝 ( 𝜒 ∧ 𝜏 ∧ 𝜂 ) ) )
36	35	bnj132	⊢ ( ∃ 𝑓 ∃ 𝑛 ∃ 𝑖 ∃ 𝑚 ∃ 𝑝 ( 𝜃 → ( 𝜒 ∧ 𝜏 ∧ 𝜂 ) ) ↔ ( 𝜃 → ∃ 𝑓 ∃ 𝑛 ∃ 𝑖 ∃ 𝑚 ∃ 𝑝 ( 𝜒 ∧ 𝜏 ∧ 𝜂 ) ) )
37	27 36	mpbir	⊢ ∃ 𝑓 ∃ 𝑛 ∃ 𝑖 ∃ 𝑚 ∃ 𝑝 ( 𝜃 → ( 𝜒 ∧ 𝜏 ∧ 𝜂 ) )