Theorem bnj1121

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	bnj1121.1	⊢ ( 𝜃 ↔ ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) )
	bnj1121.2	⊢ ( 𝜏 ↔ ( 𝐵 ∈ V ∧ TrFo ( 𝐵 , 𝐴 , 𝑅 ) ∧ pred ( 𝑋 , 𝐴 , 𝑅 ) ⊆ 𝐵 ) )
	bnj1121.3	⊢ ( 𝜒 ↔ ( 𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) )
	bnj1121.4	⊢ ( 𝜁 ↔ ( 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ ( 𝑓 ‘ 𝑖 ) ) )
	bnj1121.5	⊢ ( 𝜂 ↔ ( ( 𝑓 ∈ 𝐾 ∧ 𝑖 ∈ dom 𝑓 ) → ( 𝑓 ‘ 𝑖 ) ⊆ 𝐵 ) )
	bnj1121.6	⊢ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → ∀ 𝑖 ∈ 𝑛 𝜂 )
	bnj1121.7	⊢ 𝐾 = { 𝑓 ∣ ∃ 𝑛 ∈ 𝐷 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) }
Assertion	bnj1121	⊢ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → 𝑧 ∈ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		bnj1121.1	⊢ ( 𝜃 ↔ ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) )
2		bnj1121.2	⊢ ( 𝜏 ↔ ( 𝐵 ∈ V ∧ TrFo ( 𝐵 , 𝐴 , 𝑅 ) ∧ pred ( 𝑋 , 𝐴 , 𝑅 ) ⊆ 𝐵 ) )
3		bnj1121.3	⊢ ( 𝜒 ↔ ( 𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) )
4		bnj1121.4	⊢ ( 𝜁 ↔ ( 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ ( 𝑓 ‘ 𝑖 ) ) )
5		bnj1121.5	⊢ ( 𝜂 ↔ ( ( 𝑓 ∈ 𝐾 ∧ 𝑖 ∈ dom 𝑓 ) → ( 𝑓 ‘ 𝑖 ) ⊆ 𝐵 ) )
6		bnj1121.6	⊢ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → ∀ 𝑖 ∈ 𝑛 𝜂 )
7		bnj1121.7	⊢ 𝐾 = { 𝑓 ∣ ∃ 𝑛 ∈ 𝐷 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) }
8		19.8a	⊢ ( 𝜒 → ∃ 𝑛 𝜒 )
9	8	bnj707	⊢ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → ∃ 𝑛 𝜒 )
10	3 7	bnj1083	⊢ ( 𝑓 ∈ 𝐾 ↔ ∃ 𝑛 𝜒 )
11	9 10	sylibr	⊢ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → 𝑓 ∈ 𝐾 )
12	4	simplbi	⊢ ( 𝜁 → 𝑖 ∈ 𝑛 )
13	12	bnj708	⊢ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → 𝑖 ∈ 𝑛 )
14	3	bnj1235	⊢ ( 𝜒 → 𝑓 Fn 𝑛 )
15	14	bnj707	⊢ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → 𝑓 Fn 𝑛 )
16	15	fndmd	⊢ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → dom 𝑓 = 𝑛 )
17	13 16	eleqtrrd	⊢ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → 𝑖 ∈ dom 𝑓 )
18	6 13	bnj1294	⊢ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → 𝜂 )
19	18 5	sylib	⊢ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → ( ( 𝑓 ∈ 𝐾 ∧ 𝑖 ∈ dom 𝑓 ) → ( 𝑓 ‘ 𝑖 ) ⊆ 𝐵 ) )
20	11 17 19	mp2and	⊢ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → ( 𝑓 ‘ 𝑖 ) ⊆ 𝐵 )
21	4	simprbi	⊢ ( 𝜁 → 𝑧 ∈ ( 𝑓 ‘ 𝑖 ) )
22	21	bnj708	⊢ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → 𝑧 ∈ ( 𝑓 ‘ 𝑖 ) )
23	20 22	sseldd	⊢ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → 𝑧 ∈ 𝐵 )