Theorem bnj1445

Description: Technical lemma for bnj60 . 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	bnj1445.1	⊢ 𝐵 = { 𝑑 ∣ ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
	bnj1445.2	⊢ 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
	bnj1445.3	⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) }
	bnj1445.4	⊢ ( 𝜏 ↔ ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
	bnj1445.5	⊢ 𝐷 = { 𝑥 ∈ 𝐴 ∣ ¬ ∃ 𝑓 𝜏 }
	bnj1445.6	⊢ ( 𝜓 ↔ ( 𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅ ) )
	bnj1445.7	⊢ ( 𝜒 ↔ ( 𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀ 𝑦 ∈ 𝐷 ¬ 𝑦 𝑅 𝑥 ) )
	bnj1445.8	⊢ ( 𝜏′ ↔ [ 𝑦 / 𝑥 ] 𝜏 )
	bnj1445.9	⊢ 𝐻 = { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) 𝜏′ }
	bnj1445.10	⊢ 𝑃 = ∪ 𝐻
	bnj1445.11	⊢ 𝑍 = ⟨ 𝑥 , ( 𝑃 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
	bnj1445.12	⊢ 𝑄 = ( 𝑃 ∪ { ⟨ 𝑥 , ( 𝐺 ‘ 𝑍 ) ⟩ } )
	bnj1445.13	⊢ 𝑊 = ⟨ 𝑧 , ( 𝑄 ↾ pred ( 𝑧 , 𝐴 , 𝑅 ) ) ⟩
	bnj1445.14	⊢ 𝐸 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) )
	bnj1445.15	⊢ ( 𝜒 → 𝑃 Fn trCl ( 𝑥 , 𝐴 , 𝑅 ) )
	bnj1445.16	⊢ ( 𝜒 → 𝑄 Fn ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) )
	bnj1445.17	⊢ ( 𝜃 ↔ ( 𝜒 ∧ 𝑧 ∈ 𝐸 ) )
	bnj1445.18	⊢ ( 𝜂 ↔ ( 𝜃 ∧ 𝑧 ∈ { 𝑥 } ) )
	bnj1445.19	⊢ ( 𝜁 ↔ ( 𝜃 ∧ 𝑧 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) )
	bnj1445.20	⊢ ( 𝜌 ↔ ( 𝜁 ∧ 𝑓 ∈ 𝐻 ∧ 𝑧 ∈ dom 𝑓 ) )
	bnj1445.21	⊢ ( 𝜎 ↔ ( 𝜌 ∧ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ∧ 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑦 } ∪ trCl ( 𝑦 , 𝐴 , 𝑅 ) ) ) )
	bnj1445.22	⊢ ( 𝜑 ↔ ( 𝜎 ∧ 𝑑 ∈ 𝐵 ∧ 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) )
	bnj1445.23	⊢ 𝑋 = ⟨ 𝑧 , ( 𝑓 ↾ pred ( 𝑧 , 𝐴 , 𝑅 ) ) ⟩
Assertion	bnj1445	⊢ ( 𝜎 → ∀ 𝑑 𝜎 )

Proof

Step	Hyp	Ref	Expression
1		bnj1445.1	⊢ 𝐵 = { 𝑑 ∣ ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
2		bnj1445.2	⊢ 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
3		bnj1445.3	⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) }
4		bnj1445.4	⊢ ( 𝜏 ↔ ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
5		bnj1445.5	⊢ 𝐷 = { 𝑥 ∈ 𝐴 ∣ ¬ ∃ 𝑓 𝜏 }
6		bnj1445.6	⊢ ( 𝜓 ↔ ( 𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅ ) )
7		bnj1445.7	⊢ ( 𝜒 ↔ ( 𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀ 𝑦 ∈ 𝐷 ¬ 𝑦 𝑅 𝑥 ) )
8		bnj1445.8	⊢ ( 𝜏′ ↔ [ 𝑦 / 𝑥 ] 𝜏 )
9		bnj1445.9	⊢ 𝐻 = { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) 𝜏′ }
10		bnj1445.10	⊢ 𝑃 = ∪ 𝐻
11		bnj1445.11	⊢ 𝑍 = ⟨ 𝑥 , ( 𝑃 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
12		bnj1445.12	⊢ 𝑄 = ( 𝑃 ∪ { ⟨ 𝑥 , ( 𝐺 ‘ 𝑍 ) ⟩ } )
13		bnj1445.13	⊢ 𝑊 = ⟨ 𝑧 , ( 𝑄 ↾ pred ( 𝑧 , 𝐴 , 𝑅 ) ) ⟩
14		bnj1445.14	⊢ 𝐸 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) )
15		bnj1445.15	⊢ ( 𝜒 → 𝑃 Fn trCl ( 𝑥 , 𝐴 , 𝑅 ) )
16		bnj1445.16	⊢ ( 𝜒 → 𝑄 Fn ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) )
17		bnj1445.17	⊢ ( 𝜃 ↔ ( 𝜒 ∧ 𝑧 ∈ 𝐸 ) )
18		bnj1445.18	⊢ ( 𝜂 ↔ ( 𝜃 ∧ 𝑧 ∈ { 𝑥 } ) )
19		bnj1445.19	⊢ ( 𝜁 ↔ ( 𝜃 ∧ 𝑧 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) )
20		bnj1445.20	⊢ ( 𝜌 ↔ ( 𝜁 ∧ 𝑓 ∈ 𝐻 ∧ 𝑧 ∈ dom 𝑓 ) )
21		bnj1445.21	⊢ ( 𝜎 ↔ ( 𝜌 ∧ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ∧ 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑦 } ∪ trCl ( 𝑦 , 𝐴 , 𝑅 ) ) ) )
22		bnj1445.22	⊢ ( 𝜑 ↔ ( 𝜎 ∧ 𝑑 ∈ 𝐵 ∧ 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) )
23		bnj1445.23	⊢ 𝑋 = ⟨ 𝑧 , ( 𝑓 ↾ pred ( 𝑧 , 𝐴 , 𝑅 ) ) ⟩
24		nfv	⊢ Ⅎ 𝑑 𝑅 FrSe 𝐴
25		nfre1	⊢ Ⅎ 𝑑 ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) )
26	25	nfab	⊢ Ⅎ 𝑑 { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) }
27	3 26	nfcxfr	⊢ Ⅎ 𝑑 𝐶
28	27	nfcri	⊢ Ⅎ 𝑑 𝑓 ∈ 𝐶
29		nfv	⊢ Ⅎ 𝑑 dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) )
30	28 29	nfan	⊢ Ⅎ 𝑑 ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) )
31	4 30	nfxfr	⊢ Ⅎ 𝑑 𝜏
32	31	nfex	⊢ Ⅎ 𝑑 ∃ 𝑓 𝜏
33	32	nfn	⊢ Ⅎ 𝑑 ¬ ∃ 𝑓 𝜏
34		nfcv	⊢ Ⅎ 𝑑 𝐴
35	33 34	nfrabw	⊢ Ⅎ 𝑑 { 𝑥 ∈ 𝐴 ∣ ¬ ∃ 𝑓 𝜏 }
36	5 35	nfcxfr	⊢ Ⅎ 𝑑 𝐷
37		nfcv	⊢ Ⅎ 𝑑 ∅
38	36 37	nfne	⊢ Ⅎ 𝑑 𝐷 ≠ ∅
39	24 38	nfan	⊢ Ⅎ 𝑑 ( 𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅ )
40	6 39	nfxfr	⊢ Ⅎ 𝑑 𝜓
41	36	nfcri	⊢ Ⅎ 𝑑 𝑥 ∈ 𝐷
42		nfv	⊢ Ⅎ 𝑑 ¬ 𝑦 𝑅 𝑥
43	36 42	nfralw	⊢ Ⅎ 𝑑 ∀ 𝑦 ∈ 𝐷 ¬ 𝑦 𝑅 𝑥
44	40 41 43	nf3an	⊢ Ⅎ 𝑑 ( 𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀ 𝑦 ∈ 𝐷 ¬ 𝑦 𝑅 𝑥 )
45	7 44	nfxfr	⊢ Ⅎ 𝑑 𝜒
46	45	nf5ri	⊢ ( 𝜒 → ∀ 𝑑 𝜒 )
47	46	bnj1351	⊢ ( ( 𝜒 ∧ 𝑧 ∈ 𝐸 ) → ∀ 𝑑 ( 𝜒 ∧ 𝑧 ∈ 𝐸 ) )
48	47	nf5i	⊢ Ⅎ 𝑑 ( 𝜒 ∧ 𝑧 ∈ 𝐸 )
49	17 48	nfxfr	⊢ Ⅎ 𝑑 𝜃
50		nfv	⊢ Ⅎ 𝑑 𝑧 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 )
51	49 50	nfan	⊢ Ⅎ 𝑑 ( 𝜃 ∧ 𝑧 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) )
52	19 51	nfxfr	⊢ Ⅎ 𝑑 𝜁
53		nfcv	⊢ Ⅎ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 )
54		nfcv	⊢ Ⅎ 𝑑 𝑦
55	54 31	nfsbcw	⊢ Ⅎ 𝑑 [ 𝑦 / 𝑥 ] 𝜏
56	8 55	nfxfr	⊢ Ⅎ 𝑑 𝜏′
57	53 56	nfrexw	⊢ Ⅎ 𝑑 ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) 𝜏′
58	57	nfab	⊢ Ⅎ 𝑑 { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) 𝜏′ }
59	9 58	nfcxfr	⊢ Ⅎ 𝑑 𝐻
60	59	nfcri	⊢ Ⅎ 𝑑 𝑓 ∈ 𝐻
61		nfv	⊢ Ⅎ 𝑑 𝑧 ∈ dom 𝑓
62	52 60 61	nf3an	⊢ Ⅎ 𝑑 ( 𝜁 ∧ 𝑓 ∈ 𝐻 ∧ 𝑧 ∈ dom 𝑓 )
63	20 62	nfxfr	⊢ Ⅎ 𝑑 𝜌
64	63	nf5ri	⊢ ( 𝜌 → ∀ 𝑑 𝜌 )
65		ax-5	⊢ ( 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) → ∀ 𝑑 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) )
66	28	nf5ri	⊢ ( 𝑓 ∈ 𝐶 → ∀ 𝑑 𝑓 ∈ 𝐶 )
67		ax-5	⊢ ( dom 𝑓 = ( { 𝑦 } ∪ trCl ( 𝑦 , 𝐴 , 𝑅 ) ) → ∀ 𝑑 dom 𝑓 = ( { 𝑦 } ∪ trCl ( 𝑦 , 𝐴 , 𝑅 ) ) )
68	64 65 66 67	bnj982	⊢ ( ( 𝜌 ∧ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ∧ 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑦 } ∪ trCl ( 𝑦 , 𝐴 , 𝑅 ) ) ) → ∀ 𝑑 ( 𝜌 ∧ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) ∧ 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑦 } ∪ trCl ( 𝑦 , 𝐴 , 𝑅 ) ) ) )
69	21 68	hbxfrbi	⊢ ( 𝜎 → ∀ 𝑑 𝜎 )