bnj1033

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	bnj1033.1	⊢ ( 𝜑 ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) )
	bnj1033.2	⊢ ( 𝜓 ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
	bnj1033.3	⊢ ( 𝜒 ↔ ( 𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) )
	bnj1033.4	⊢ ( 𝜃 ↔ ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) )
	bnj1033.5	⊢ ( 𝜏 ↔ ( 𝐵 ∈ V ∧ TrFo ( 𝐵 , 𝐴 , 𝑅 ) ∧ pred ( 𝑋 , 𝐴 , 𝑅 ) ⊆ 𝐵 ) )
	bnj1033.6	⊢ ( 𝜂 ↔ 𝑧 ∈ trCl ( 𝑋 , 𝐴 , 𝑅 ) )
	bnj1033.7	⊢ ( 𝜁 ↔ ( 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ ( 𝑓 ‘ 𝑖 ) ) )
	bnj1033.8	⊢ 𝐷 = ( ω ∖ { ∅ } )
	bnj1033.9	⊢ 𝐾 = { 𝑓 ∣ ∃ 𝑛 ∈ 𝐷 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) }
	bnj1033.10	⊢ ( ∃ 𝑓 ∃ 𝑛 ∃ 𝑖 ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → 𝑧 ∈ 𝐵 )
Assertion	bnj1033	⊢ ( ( 𝜃 ∧ 𝜏 ) → trCl ( 𝑋 , 𝐴 , 𝑅 ) ⊆ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		bnj1033.1	⊢ ( 𝜑 ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) )
2		bnj1033.2	⊢ ( 𝜓 ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) )
3		bnj1033.3	⊢ ( 𝜒 ↔ ( 𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) )
4		bnj1033.4	⊢ ( 𝜃 ↔ ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) )
5		bnj1033.5	⊢ ( 𝜏 ↔ ( 𝐵 ∈ V ∧ TrFo ( 𝐵 , 𝐴 , 𝑅 ) ∧ pred ( 𝑋 , 𝐴 , 𝑅 ) ⊆ 𝐵 ) )
6		bnj1033.6	⊢ ( 𝜂 ↔ 𝑧 ∈ trCl ( 𝑋 , 𝐴 , 𝑅 ) )
7		bnj1033.7	⊢ ( 𝜁 ↔ ( 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ ( 𝑓 ‘ 𝑖 ) ) )
8		bnj1033.8	⊢ 𝐷 = ( ω ∖ { ∅ } )
9		bnj1033.9	⊢ 𝐾 = { 𝑓 ∣ ∃ 𝑛 ∈ 𝐷 ( 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) }
10		bnj1033.10	⊢ ( ∃ 𝑓 ∃ 𝑛 ∃ 𝑖 ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → 𝑧 ∈ 𝐵 )
11	1 2 8 9 3	bnj983	⊢ ( 𝑧 ∈ trCl ( 𝑋 , 𝐴 , 𝑅 ) ↔ ∃ 𝑓 ∃ 𝑛 ∃ 𝑖 ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ ( 𝑓 ‘ 𝑖 ) ) )
12		19.42v	⊢ ( ∃ 𝑖 ( ( 𝜃 ∧ 𝜏 ) ∧ ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ ( 𝑓 ‘ 𝑖 ) ) ) ↔ ( ( 𝜃 ∧ 𝜏 ) ∧ ∃ 𝑖 ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ ( 𝑓 ‘ 𝑖 ) ) ) )
13		df-3an	⊢ ( ( 𝜃 ∧ 𝜏 ∧ ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ ( 𝑓 ‘ 𝑖 ) ) ) ↔ ( ( 𝜃 ∧ 𝜏 ) ∧ ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ ( 𝑓 ‘ 𝑖 ) ) ) )
14	13	exbii	⊢ ( ∃ 𝑖 ( 𝜃 ∧ 𝜏 ∧ ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ ( 𝑓 ‘ 𝑖 ) ) ) ↔ ∃ 𝑖 ( ( 𝜃 ∧ 𝜏 ) ∧ ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ ( 𝑓 ‘ 𝑖 ) ) ) )
15		df-3an	⊢ ( ( 𝜃 ∧ 𝜏 ∧ ∃ 𝑖 ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ ( 𝑓 ‘ 𝑖 ) ) ) ↔ ( ( 𝜃 ∧ 𝜏 ) ∧ ∃ 𝑖 ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ ( 𝑓 ‘ 𝑖 ) ) ) )
16	12 14 15	3bitr4i	⊢ ( ∃ 𝑖 ( 𝜃 ∧ 𝜏 ∧ ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ ( 𝑓 ‘ 𝑖 ) ) ) ↔ ( 𝜃 ∧ 𝜏 ∧ ∃ 𝑖 ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ ( 𝑓 ‘ 𝑖 ) ) ) )
17	16	exbii	⊢ ( ∃ 𝑛 ∃ 𝑖 ( 𝜃 ∧ 𝜏 ∧ ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ ( 𝑓 ‘ 𝑖 ) ) ) ↔ ∃ 𝑛 ( 𝜃 ∧ 𝜏 ∧ ∃ 𝑖 ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ ( 𝑓 ‘ 𝑖 ) ) ) )
18		19.42v	⊢ ( ∃ 𝑛 ( ( 𝜃 ∧ 𝜏 ) ∧ ∃ 𝑖 ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ ( 𝑓 ‘ 𝑖 ) ) ) ↔ ( ( 𝜃 ∧ 𝜏 ) ∧ ∃ 𝑛 ∃ 𝑖 ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ ( 𝑓 ‘ 𝑖 ) ) ) )
19	15	exbii	⊢ ( ∃ 𝑛 ( 𝜃 ∧ 𝜏 ∧ ∃ 𝑖 ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ ( 𝑓 ‘ 𝑖 ) ) ) ↔ ∃ 𝑛 ( ( 𝜃 ∧ 𝜏 ) ∧ ∃ 𝑖 ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ ( 𝑓 ‘ 𝑖 ) ) ) )
20		df-3an	⊢ ( ( 𝜃 ∧ 𝜏 ∧ ∃ 𝑛 ∃ 𝑖 ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ ( 𝑓 ‘ 𝑖 ) ) ) ↔ ( ( 𝜃 ∧ 𝜏 ) ∧ ∃ 𝑛 ∃ 𝑖 ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ ( 𝑓 ‘ 𝑖 ) ) ) )
21	18 19 20	3bitr4i	⊢ ( ∃ 𝑛 ( 𝜃 ∧ 𝜏 ∧ ∃ 𝑖 ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ ( 𝑓 ‘ 𝑖 ) ) ) ↔ ( 𝜃 ∧ 𝜏 ∧ ∃ 𝑛 ∃ 𝑖 ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ ( 𝑓 ‘ 𝑖 ) ) ) )
22	17 21	bitri	⊢ ( ∃ 𝑛 ∃ 𝑖 ( 𝜃 ∧ 𝜏 ∧ ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ ( 𝑓 ‘ 𝑖 ) ) ) ↔ ( 𝜃 ∧ 𝜏 ∧ ∃ 𝑛 ∃ 𝑖 ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ ( 𝑓 ‘ 𝑖 ) ) ) )
23	22	exbii	⊢ ( ∃ 𝑓 ∃ 𝑛 ∃ 𝑖 ( 𝜃 ∧ 𝜏 ∧ ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ ( 𝑓 ‘ 𝑖 ) ) ) ↔ ∃ 𝑓 ( 𝜃 ∧ 𝜏 ∧ ∃ 𝑛 ∃ 𝑖 ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ ( 𝑓 ‘ 𝑖 ) ) ) )
24		19.42v	⊢ ( ∃ 𝑓 ( ( 𝜃 ∧ 𝜏 ) ∧ ∃ 𝑛 ∃ 𝑖 ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ ( 𝑓 ‘ 𝑖 ) ) ) ↔ ( ( 𝜃 ∧ 𝜏 ) ∧ ∃ 𝑓 ∃ 𝑛 ∃ 𝑖 ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ ( 𝑓 ‘ 𝑖 ) ) ) )
25	20	exbii	⊢ ( ∃ 𝑓 ( 𝜃 ∧ 𝜏 ∧ ∃ 𝑛 ∃ 𝑖 ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ ( 𝑓 ‘ 𝑖 ) ) ) ↔ ∃ 𝑓 ( ( 𝜃 ∧ 𝜏 ) ∧ ∃ 𝑛 ∃ 𝑖 ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ ( 𝑓 ‘ 𝑖 ) ) ) )
26		df-3an	⊢ ( ( 𝜃 ∧ 𝜏 ∧ ∃ 𝑓 ∃ 𝑛 ∃ 𝑖 ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ ( 𝑓 ‘ 𝑖 ) ) ) ↔ ( ( 𝜃 ∧ 𝜏 ) ∧ ∃ 𝑓 ∃ 𝑛 ∃ 𝑖 ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ ( 𝑓 ‘ 𝑖 ) ) ) )
27	24 25 26	3bitr4i	⊢ ( ∃ 𝑓 ( 𝜃 ∧ 𝜏 ∧ ∃ 𝑛 ∃ 𝑖 ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ ( 𝑓 ‘ 𝑖 ) ) ) ↔ ( 𝜃 ∧ 𝜏 ∧ ∃ 𝑓 ∃ 𝑛 ∃ 𝑖 ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ ( 𝑓 ‘ 𝑖 ) ) ) )
28	23 27	bitri	⊢ ( ∃ 𝑓 ∃ 𝑛 ∃ 𝑖 ( 𝜃 ∧ 𝜏 ∧ ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ ( 𝑓 ‘ 𝑖 ) ) ) ↔ ( 𝜃 ∧ 𝜏 ∧ ∃ 𝑓 ∃ 𝑛 ∃ 𝑖 ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ ( 𝑓 ‘ 𝑖 ) ) ) )
29		bnj255	⊢ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) ↔ ( 𝜃 ∧ 𝜏 ∧ ( 𝜒 ∧ 𝜁 ) ) )
30	7	anbi2i	⊢ ( ( 𝜒 ∧ 𝜁 ) ↔ ( 𝜒 ∧ ( 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ ( 𝑓 ‘ 𝑖 ) ) ) )
31		3anass	⊢ ( ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ ( 𝑓 ‘ 𝑖 ) ) ↔ ( 𝜒 ∧ ( 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ ( 𝑓 ‘ 𝑖 ) ) ) )
32	30 31	bitr4i	⊢ ( ( 𝜒 ∧ 𝜁 ) ↔ ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ ( 𝑓 ‘ 𝑖 ) ) )
33	32	3anbi3i	⊢ ( ( 𝜃 ∧ 𝜏 ∧ ( 𝜒 ∧ 𝜁 ) ) ↔ ( 𝜃 ∧ 𝜏 ∧ ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ ( 𝑓 ‘ 𝑖 ) ) ) )
34	29 33	bitri	⊢ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) ↔ ( 𝜃 ∧ 𝜏 ∧ ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ ( 𝑓 ‘ 𝑖 ) ) ) )
35	34	3exbii	⊢ ( ∃ 𝑓 ∃ 𝑛 ∃ 𝑖 ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) ↔ ∃ 𝑓 ∃ 𝑛 ∃ 𝑖 ( 𝜃 ∧ 𝜏 ∧ ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ ( 𝑓 ‘ 𝑖 ) ) ) )
36	35 10	sylbir	⊢ ( ∃ 𝑓 ∃ 𝑛 ∃ 𝑖 ( 𝜃 ∧ 𝜏 ∧ ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ ( 𝑓 ‘ 𝑖 ) ) ) → 𝑧 ∈ 𝐵 )
37	28 36	sylbir	⊢ ( ( 𝜃 ∧ 𝜏 ∧ ∃ 𝑓 ∃ 𝑛 ∃ 𝑖 ( 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ ( 𝑓 ‘ 𝑖 ) ) ) → 𝑧 ∈ 𝐵 )
38	11 37	syl3an3b	⊢ ( ( 𝜃 ∧ 𝜏 ∧ 𝑧 ∈ trCl ( 𝑋 , 𝐴 , 𝑅 ) ) → 𝑧 ∈ 𝐵 )
39	38	3expia	⊢ ( ( 𝜃 ∧ 𝜏 ) → ( 𝑧 ∈ trCl ( 𝑋 , 𝐴 , 𝑅 ) → 𝑧 ∈ 𝐵 ) )
40	39	ssrdv	⊢ ( ( 𝜃 ∧ 𝜏 ) → trCl ( 𝑋 , 𝐴 , 𝑅 ) ⊆ 𝐵 )

Metamath Proof Explorer

Theorem bnj1033

Proof