Theorem bnj1052

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	bnj1052.1	⊢ ( 𝜑 ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) )
	bnj1052.2	⊢ ( 𝜓 ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
	bnj1052.3	⊢ ( 𝜒 ↔ ( 𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) )
	bnj1052.4	⊢ ( 𝜃 ↔ ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) )
	bnj1052.5	⊢ ( 𝜏 ↔ ( 𝐵 ∈ V ∧ TrFo ( 𝐵 , 𝐴 , 𝑅 ) ∧ pred ( 𝑋 , 𝐴 , 𝑅 ) ⊆ 𝐵 ) )
	bnj1052.6	⊢ ( 𝜁 ↔ ( 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ ( 𝑓 ‘ 𝑖 ) ) )
	bnj1052.7	⊢ 𝐷 = ( ω ∖ { ∅ } )
	bnj1052.8	⊢ 𝐾 = { 𝑓 ∣ ∃ 𝑛 ∈ 𝐷 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) }
	bnj1052.9	⊢ ( 𝜂 ↔ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → 𝑧 ∈ 𝐵 ) )
	bnj1052.10	⊢ ( 𝜌 ↔ ∀ 𝑗 ∈ 𝑛 ( 𝑗 E 𝑖 → [ 𝑗 / 𝑖 ] 𝜂 ) )
	bnj1052.37	⊢ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → ( E Fr 𝑛 ∧ ∀ 𝑖 ∈ 𝑛 ( 𝜌 → 𝜂 ) ) )
Assertion	bnj1052	⊢ ( ( 𝜃 ∧ 𝜏 ) → trCl ( 𝑋 , 𝐴 , 𝑅 ) ⊆ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		bnj1052.1	⊢ ( 𝜑 ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) )
2		bnj1052.2	⊢ ( 𝜓 ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
3		bnj1052.3	⊢ ( 𝜒 ↔ ( 𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) )
4		bnj1052.4	⊢ ( 𝜃 ↔ ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) )
5		bnj1052.5	⊢ ( 𝜏 ↔ ( 𝐵 ∈ V ∧ TrFo ( 𝐵 , 𝐴 , 𝑅 ) ∧ pred ( 𝑋 , 𝐴 , 𝑅 ) ⊆ 𝐵 ) )
6		bnj1052.6	⊢ ( 𝜁 ↔ ( 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ ( 𝑓 ‘ 𝑖 ) ) )
7		bnj1052.7	⊢ 𝐷 = ( ω ∖ { ∅ } )
8		bnj1052.8	⊢ 𝐾 = { 𝑓 ∣ ∃ 𝑛 ∈ 𝐷 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) }
9		bnj1052.9	⊢ ( 𝜂 ↔ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → 𝑧 ∈ 𝐵 ) )
10		bnj1052.10	⊢ ( 𝜌 ↔ ∀ 𝑗 ∈ 𝑛 ( 𝑗 E 𝑖 → [ 𝑗 / 𝑖 ] 𝜂 ) )
11		bnj1052.37	⊢ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → ( E Fr 𝑛 ∧ ∀ 𝑖 ∈ 𝑛 ( 𝜌 → 𝜂 ) ) )
12		19.23vv	⊢ ( ∀ 𝑛 ∀ 𝑖 ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → 𝑧 ∈ 𝐵 ) ↔ ( ∃ 𝑛 ∃ 𝑖 ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → 𝑧 ∈ 𝐵 ) )
13	12	albii	⊢ ( ∀ 𝑓 ∀ 𝑛 ∀ 𝑖 ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → 𝑧 ∈ 𝐵 ) ↔ ∀ 𝑓 ( ∃ 𝑛 ∃ 𝑖 ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → 𝑧 ∈ 𝐵 ) )
14		19.23v	⊢ ( ∀ 𝑓 ( ∃ 𝑛 ∃ 𝑖 ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → 𝑧 ∈ 𝐵 ) ↔ ( ∃ 𝑓 ∃ 𝑛 ∃ 𝑖 ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → 𝑧 ∈ 𝐵 ) )
15	13 14	bitri	⊢ ( ∀ 𝑓 ∀ 𝑛 ∀ 𝑖 ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → 𝑧 ∈ 𝐵 ) ↔ ( ∃ 𝑓 ∃ 𝑛 ∃ 𝑖 ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → 𝑧 ∈ 𝐵 ) )
16		vex	⊢ 𝑛 ∈ V
17	16 10	bnj110	⊢ ( ( E Fr 𝑛 ∧ ∀ 𝑖 ∈ 𝑛 ( 𝜌 → 𝜂 ) ) → ∀ 𝑖 ∈ 𝑛 𝜂 )
18	6 9	bnj1049	⊢ ( ∀ 𝑖 ∈ 𝑛 𝜂 ↔ ∀ 𝑖 𝜂 )
19	17 18	sylib	⊢ ( ( E Fr 𝑛 ∧ ∀ 𝑖 ∈ 𝑛 ( 𝜌 → 𝜂 ) ) → ∀ 𝑖 𝜂 )
20	19	19.21bi	⊢ ( ( E Fr 𝑛 ∧ ∀ 𝑖 ∈ 𝑛 ( 𝜌 → 𝜂 ) ) → 𝜂 )
21	20 9	sylib	⊢ ( ( E Fr 𝑛 ∧ ∀ 𝑖 ∈ 𝑛 ( 𝜌 → 𝜂 ) ) → ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → 𝑧 ∈ 𝐵 ) )
22	11 21	mpcom	⊢ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → 𝑧 ∈ 𝐵 )
23	22	gen2	⊢ ∀ 𝑛 ∀ 𝑖 ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → 𝑧 ∈ 𝐵 )
24	15 23	mpgbi	⊢ ( ∃ 𝑓 ∃ 𝑛 ∃ 𝑖 ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → 𝑧 ∈ 𝐵 )
25	1 2 3 4 5 6 7 8 24	bnj1034	⊢ ( ( 𝜃 ∧ 𝜏 ) → trCl ( 𝑋 , 𝐴 , 𝑅 ) ⊆ 𝐵 )