Theorem bnj1030

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	bnj1030.1	⊢ ( 𝜑 ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) )
	bnj1030.2	⊢ ( 𝜓 ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
	bnj1030.3	⊢ ( 𝜒 ↔ ( 𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) )
	bnj1030.4	⊢ ( 𝜃 ↔ ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) )
	bnj1030.5	⊢ ( 𝜏 ↔ ( 𝐵 ∈ V ∧ TrFo ( 𝐵 , 𝐴 , 𝑅 ) ∧ pred ( 𝑋 , 𝐴 , 𝑅 ) ⊆ 𝐵 ) )
	bnj1030.6	⊢ ( 𝜁 ↔ ( 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ ( 𝑓 ‘ 𝑖 ) ) )
	bnj1030.7	⊢ 𝐷 = ( ω ∖ { ∅ } )
	bnj1030.8	⊢ 𝐾 = { 𝑓 ∣ ∃ 𝑛 ∈ 𝐷 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) }
	bnj1030.9	⊢ ( 𝜂 ↔ ( ( 𝑓 ∈ 𝐾 ∧ 𝑖 ∈ dom 𝑓 ) → ( 𝑓 ‘ 𝑖 ) ⊆ 𝐵 ) )
	bnj1030.10	⊢ ( 𝜌 ↔ ∀ 𝑗 ∈ 𝑛 ( 𝑗 E 𝑖 → [ 𝑗 / 𝑖 ] 𝜂 ) )
	bnj1030.11	⊢ ( 𝜑′ ↔ [ 𝑗 / 𝑖 ] 𝜑 )
	bnj1030.12	⊢ ( 𝜓′ ↔ [ 𝑗 / 𝑖 ] 𝜓 )
	bnj1030.13	⊢ ( 𝜒′ ↔ [ 𝑗 / 𝑖 ] 𝜒 )
	bnj1030.14	⊢ ( 𝜃′ ↔ [ 𝑗 / 𝑖 ] 𝜃 )
	bnj1030.15	⊢ ( 𝜏′ ↔ [ 𝑗 / 𝑖 ] 𝜏 )
	bnj1030.16	⊢ ( 𝜁′ ↔ [ 𝑗 / 𝑖 ] 𝜁 )
	bnj1030.17	⊢ ( 𝜂′ ↔ [ 𝑗 / 𝑖 ] 𝜂 )
	bnj1030.18	⊢ ( 𝜎 ↔ ( ( 𝑗 ∈ 𝑛 ∧ 𝑗 E 𝑖 ) → 𝜂′ ) )
	bnj1030.19	⊢ ( 𝜑0 ↔ ( 𝑖 ∈ 𝑛 ∧ 𝜎 ∧ 𝑓 ∈ 𝐾 ∧ 𝑖 ∈ dom 𝑓 ) )
Assertion	bnj1030	⊢ ( ( 𝜃 ∧ 𝜏 ) → trCl ( 𝑋 , 𝐴 , 𝑅 ) ⊆ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		bnj1030.1	⊢ ( 𝜑 ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) )
2		bnj1030.2	⊢ ( 𝜓 ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
3		bnj1030.3	⊢ ( 𝜒 ↔ ( 𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) )
4		bnj1030.4	⊢ ( 𝜃 ↔ ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) )
5		bnj1030.5	⊢ ( 𝜏 ↔ ( 𝐵 ∈ V ∧ TrFo ( 𝐵 , 𝐴 , 𝑅 ) ∧ pred ( 𝑋 , 𝐴 , 𝑅 ) ⊆ 𝐵 ) )
6		bnj1030.6	⊢ ( 𝜁 ↔ ( 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ ( 𝑓 ‘ 𝑖 ) ) )
7		bnj1030.7	⊢ 𝐷 = ( ω ∖ { ∅ } )
8		bnj1030.8	⊢ 𝐾 = { 𝑓 ∣ ∃ 𝑛 ∈ 𝐷 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) }
9		bnj1030.9	⊢ ( 𝜂 ↔ ( ( 𝑓 ∈ 𝐾 ∧ 𝑖 ∈ dom 𝑓 ) → ( 𝑓 ‘ 𝑖 ) ⊆ 𝐵 ) )
10		bnj1030.10	⊢ ( 𝜌 ↔ ∀ 𝑗 ∈ 𝑛 ( 𝑗 E 𝑖 → [ 𝑗 / 𝑖 ] 𝜂 ) )
11		bnj1030.11	⊢ ( 𝜑′ ↔ [ 𝑗 / 𝑖 ] 𝜑 )
12		bnj1030.12	⊢ ( 𝜓′ ↔ [ 𝑗 / 𝑖 ] 𝜓 )
13		bnj1030.13	⊢ ( 𝜒′ ↔ [ 𝑗 / 𝑖 ] 𝜒 )
14		bnj1030.14	⊢ ( 𝜃′ ↔ [ 𝑗 / 𝑖 ] 𝜃 )
15		bnj1030.15	⊢ ( 𝜏′ ↔ [ 𝑗 / 𝑖 ] 𝜏 )
16		bnj1030.16	⊢ ( 𝜁′ ↔ [ 𝑗 / 𝑖 ] 𝜁 )
17		bnj1030.17	⊢ ( 𝜂′ ↔ [ 𝑗 / 𝑖 ] 𝜂 )
18		bnj1030.18	⊢ ( 𝜎 ↔ ( ( 𝑗 ∈ 𝑛 ∧ 𝑗 E 𝑖 ) → 𝜂′ ) )
19		bnj1030.19	⊢ ( 𝜑0 ↔ ( 𝑖 ∈ 𝑛 ∧ 𝜎 ∧ 𝑓 ∈ 𝐾 ∧ 𝑖 ∈ dom 𝑓 ) )
20		19.23vv	⊢ ( ∀ 𝑛 ∀ 𝑖 ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → 𝑧 ∈ 𝐵 ) ↔ ( ∃ 𝑛 ∃ 𝑖 ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → 𝑧 ∈ 𝐵 ) )
21	20	albii	⊢ ( ∀ 𝑓 ∀ 𝑛 ∀ 𝑖 ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → 𝑧 ∈ 𝐵 ) ↔ ∀ 𝑓 ( ∃ 𝑛 ∃ 𝑖 ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → 𝑧 ∈ 𝐵 ) )
22		19.23v	⊢ ( ∀ 𝑓 ( ∃ 𝑛 ∃ 𝑖 ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → 𝑧 ∈ 𝐵 ) ↔ ( ∃ 𝑓 ∃ 𝑛 ∃ 𝑖 ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → 𝑧 ∈ 𝐵 ) )
23	21 22	bitri	⊢ ( ∀ 𝑓 ∀ 𝑛 ∀ 𝑖 ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → 𝑧 ∈ 𝐵 ) ↔ ( ∃ 𝑓 ∃ 𝑛 ∃ 𝑖 ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → 𝑧 ∈ 𝐵 ) )
24	7	bnj1071	⊢ ( 𝑛 ∈ 𝐷 → E Fr 𝑛 )
25	3 24	bnj769	⊢ ( 𝜒 → E Fr 𝑛 )
26	25	bnj707	⊢ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → E Fr 𝑛 )
27	2 8 9 17	bnj1123	⊢ ( 𝜂′ ↔ ( ( 𝑓 ∈ 𝐾 ∧ 𝑗 ∈ dom 𝑓 ) → ( 𝑓 ‘ 𝑗 ) ⊆ 𝐵 ) )
28	2 3 5 7 18 19 27	bnj1118	⊢ ∃ 𝑗 ( ( 𝑖 ≠ ∅ ∧ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ) ∧ 𝜑0 ) ) → ( 𝑓 ‘ 𝑖 ) ⊆ 𝐵 )
29	1 3 5	bnj1097	⊢ ( ( 𝑖 = ∅ ∧ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ) ∧ 𝜑0 ) ) → ( 𝑓 ‘ 𝑖 ) ⊆ 𝐵 )
30	28 29	bnj1109	⊢ ∃ 𝑗 ( ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ) ∧ 𝜑0 ) → ( 𝑓 ‘ 𝑖 ) ⊆ 𝐵 )
31	30 2 3	bnj1093	⊢ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → ∀ 𝑖 ∃ 𝑗 ( 𝜑0 → ( 𝑓 ‘ 𝑖 ) ⊆ 𝐵 ) )
32	9 10 17 18 19 31	bnj1090	⊢ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → ∀ 𝑖 ∈ 𝑛 ( 𝜌 → 𝜂 ) )
33		vex	⊢ 𝑛 ∈ V
34	33 10	bnj110	⊢ ( ( E Fr 𝑛 ∧ ∀ 𝑖 ∈ 𝑛 ( 𝜌 → 𝜂 ) ) → ∀ 𝑖 ∈ 𝑛 𝜂 )
35	26 32 34	syl2anc	⊢ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → ∀ 𝑖 ∈ 𝑛 𝜂 )
36	4 5 3 6 9 35 8	bnj1121	⊢ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → 𝑧 ∈ 𝐵 )
37	36	gen2	⊢ ∀ 𝑛 ∀ 𝑖 ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → 𝑧 ∈ 𝐵 )
38	23 37	mpgbi	⊢ ( ∃ 𝑓 ∃ 𝑛 ∃ 𝑖 ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → 𝑧 ∈ 𝐵 )
39	1 2 3 4 5 6 7 8 38	bnj1034	⊢ ( ( 𝜃 ∧ 𝜏 ) → trCl ( 𝑋 , 𝐴 , 𝑅 ) ⊆ 𝐵 )