Theorem bnj1053

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	bnj1053.1	⊢ ( 𝜑 ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) )
	bnj1053.2	⊢ ( 𝜓 ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
	bnj1053.3	⊢ ( 𝜒 ↔ ( 𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) )
	bnj1053.4	⊢ ( 𝜃 ↔ ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) )
	bnj1053.5	⊢ ( 𝜏 ↔ ( 𝐵 ∈ V ∧ TrFo ( 𝐵 , 𝐴 , 𝑅 ) ∧ pred ( 𝑋 , 𝐴 , 𝑅 ) ⊆ 𝐵 ) )
	bnj1053.6	⊢ ( 𝜁 ↔ ( 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ ( 𝑓 ‘ 𝑖 ) ) )
	bnj1053.7	⊢ 𝐷 = ( ω ∖ { ∅ } )
	bnj1053.8	⊢ 𝐾 = { 𝑓 ∣ ∃ 𝑛 ∈ 𝐷 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) }
	bnj1053.9	⊢ ( 𝜂 ↔ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → 𝑧 ∈ 𝐵 ) )
	bnj1053.10	⊢ ( 𝜌 ↔ ∀ 𝑗 ∈ 𝑛 ( 𝑗 E 𝑖 → [ 𝑗 / 𝑖 ] 𝜂 ) )
	bnj1053.37	⊢ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → ∀ 𝑖 ∈ 𝑛 ( 𝜌 → 𝜂 ) )
Assertion	bnj1053	⊢ ( ( 𝜃 ∧ 𝜏 ) → trCl ( 𝑋 , 𝐴 , 𝑅 ) ⊆ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		bnj1053.1	⊢ ( 𝜑 ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) )
2		bnj1053.2	⊢ ( 𝜓 ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
3		bnj1053.3	⊢ ( 𝜒 ↔ ( 𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) )
4		bnj1053.4	⊢ ( 𝜃 ↔ ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) )
5		bnj1053.5	⊢ ( 𝜏 ↔ ( 𝐵 ∈ V ∧ TrFo ( 𝐵 , 𝐴 , 𝑅 ) ∧ pred ( 𝑋 , 𝐴 , 𝑅 ) ⊆ 𝐵 ) )
6		bnj1053.6	⊢ ( 𝜁 ↔ ( 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ ( 𝑓 ‘ 𝑖 ) ) )
7		bnj1053.7	⊢ 𝐷 = ( ω ∖ { ∅ } )
8		bnj1053.8	⊢ 𝐾 = { 𝑓 ∣ ∃ 𝑛 ∈ 𝐷 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) }
9		bnj1053.9	⊢ ( 𝜂 ↔ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → 𝑧 ∈ 𝐵 ) )
10		bnj1053.10	⊢ ( 𝜌 ↔ ∀ 𝑗 ∈ 𝑛 ( 𝑗 E 𝑖 → [ 𝑗 / 𝑖 ] 𝜂 ) )
11		bnj1053.37	⊢ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → ∀ 𝑖 ∈ 𝑛 ( 𝜌 → 𝜂 ) )
12	7	bnj923	⊢ ( 𝑛 ∈ 𝐷 → 𝑛 ∈ ω )
13		nnord	⊢ ( 𝑛 ∈ ω → Ord 𝑛 )
14		ordfr	⊢ ( Ord 𝑛 → E Fr 𝑛 )
15	12 13 14	3syl	⊢ ( 𝑛 ∈ 𝐷 → E Fr 𝑛 )
16	3 15	bnj769	⊢ ( 𝜒 → E Fr 𝑛 )
17	16	bnj707	⊢ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → E Fr 𝑛 )
18	17 11	jca	⊢ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → ( E Fr 𝑛 ∧ ∀ 𝑖 ∈ 𝑛 ( 𝜌 → 𝜂 ) ) )
19	1 2 3 4 5 6 7 8 9 10 18	bnj1052	⊢ ( ( 𝜃 ∧ 𝜏 ) → trCl ( 𝑋 , 𝐴 , 𝑅 ) ⊆ 𝐵 )