Theorem bnj1449

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	bnj1449.1	⊢ 𝐵 = { 𝑑 ∣ ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
	bnj1449.2	⊢ 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
	bnj1449.3	⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) }
	bnj1449.4	⊢ ( 𝜏 ↔ ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
	bnj1449.5	⊢ 𝐷 = { 𝑥 ∈ 𝐴 ∣ ¬ ∃ 𝑓 𝜏 }
	bnj1449.6	⊢ ( 𝜓 ↔ ( 𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅ ) )
	bnj1449.7	⊢ ( 𝜒 ↔ ( 𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀ 𝑦 ∈ 𝐷 ¬ 𝑦 𝑅 𝑥 ) )
	bnj1449.8	⊢ ( 𝜏′ ↔ [ 𝑦 / 𝑥 ] 𝜏 )
	bnj1449.9	⊢ 𝐻 = { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) 𝜏′ }
	bnj1449.10	⊢ 𝑃 = ∪ 𝐻
	bnj1449.11	⊢ 𝑍 = ⟨ 𝑥 , ( 𝑃 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
	bnj1449.12	⊢ 𝑄 = ( 𝑃 ∪ { ⟨ 𝑥 , ( 𝐺 ‘ 𝑍 ) ⟩ } )
	bnj1449.13	⊢ 𝑊 = ⟨ 𝑧 , ( 𝑄 ↾ pred ( 𝑧 , 𝐴 , 𝑅 ) ) ⟩
	bnj1449.14	⊢ 𝐸 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) )
	bnj1449.15	⊢ ( 𝜒 → 𝑃 Fn trCl ( 𝑥 , 𝐴 , 𝑅 ) )
	bnj1449.16	⊢ ( 𝜒 → 𝑄 Fn ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) )
	bnj1449.17	⊢ ( 𝜃 ↔ ( 𝜒 ∧ 𝑧 ∈ 𝐸 ) )
	bnj1449.18	⊢ ( 𝜂 ↔ ( 𝜃 ∧ 𝑧 ∈ { 𝑥 } ) )
	bnj1449.19	⊢ ( 𝜁 ↔ ( 𝜃 ∧ 𝑧 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) )
Assertion	bnj1449	⊢ ( 𝜁 → ∀ 𝑓 𝜁 )

Proof

Step	Hyp	Ref	Expression
1		bnj1449.1	⊢ 𝐵 = { 𝑑 ∣ ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
2		bnj1449.2	⊢ 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
3		bnj1449.3	⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) }
4		bnj1449.4	⊢ ( 𝜏 ↔ ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
5		bnj1449.5	⊢ 𝐷 = { 𝑥 ∈ 𝐴 ∣ ¬ ∃ 𝑓 𝜏 }
6		bnj1449.6	⊢ ( 𝜓 ↔ ( 𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅ ) )
7		bnj1449.7	⊢ ( 𝜒 ↔ ( 𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀ 𝑦 ∈ 𝐷 ¬ 𝑦 𝑅 𝑥 ) )
8		bnj1449.8	⊢ ( 𝜏′ ↔ [ 𝑦 / 𝑥 ] 𝜏 )
9		bnj1449.9	⊢ 𝐻 = { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) 𝜏′ }
10		bnj1449.10	⊢ 𝑃 = ∪ 𝐻
11		bnj1449.11	⊢ 𝑍 = ⟨ 𝑥 , ( 𝑃 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
12		bnj1449.12	⊢ 𝑄 = ( 𝑃 ∪ { ⟨ 𝑥 , ( 𝐺 ‘ 𝑍 ) ⟩ } )
13		bnj1449.13	⊢ 𝑊 = ⟨ 𝑧 , ( 𝑄 ↾ pred ( 𝑧 , 𝐴 , 𝑅 ) ) ⟩
14		bnj1449.14	⊢ 𝐸 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) )
15		bnj1449.15	⊢ ( 𝜒 → 𝑃 Fn trCl ( 𝑥 , 𝐴 , 𝑅 ) )
16		bnj1449.16	⊢ ( 𝜒 → 𝑄 Fn ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) )
17		bnj1449.17	⊢ ( 𝜃 ↔ ( 𝜒 ∧ 𝑧 ∈ 𝐸 ) )
18		bnj1449.18	⊢ ( 𝜂 ↔ ( 𝜃 ∧ 𝑧 ∈ { 𝑥 } ) )
19		bnj1449.19	⊢ ( 𝜁 ↔ ( 𝜃 ∧ 𝑧 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) )
20		nfv	⊢ Ⅎ 𝑓 𝑅 FrSe 𝐴
21		nfe1	⊢ Ⅎ 𝑓 ∃ 𝑓 𝜏
22	21	nfn	⊢ Ⅎ 𝑓 ¬ ∃ 𝑓 𝜏
23		nfcv	⊢ Ⅎ 𝑓 𝐴
24	22 23	nfrabw	⊢ Ⅎ 𝑓 { 𝑥 ∈ 𝐴 ∣ ¬ ∃ 𝑓 𝜏 }
25	5 24	nfcxfr	⊢ Ⅎ 𝑓 𝐷
26		nfcv	⊢ Ⅎ 𝑓 ∅
27	25 26	nfne	⊢ Ⅎ 𝑓 𝐷 ≠ ∅
28	20 27	nfan	⊢ Ⅎ 𝑓 ( 𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅ )
29	6 28	nfxfr	⊢ Ⅎ 𝑓 𝜓
30	25	nfcri	⊢ Ⅎ 𝑓 𝑥 ∈ 𝐷
31		nfv	⊢ Ⅎ 𝑓 ¬ 𝑦 𝑅 𝑥
32	25 31	nfralw	⊢ Ⅎ 𝑓 ∀ 𝑦 ∈ 𝐷 ¬ 𝑦 𝑅 𝑥
33	29 30 32	nf3an	⊢ Ⅎ 𝑓 ( 𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀ 𝑦 ∈ 𝐷 ¬ 𝑦 𝑅 𝑥 )
34	7 33	nfxfr	⊢ Ⅎ 𝑓 𝜒
35		nfv	⊢ Ⅎ 𝑓 𝑧 ∈ 𝐸
36	34 35	nfan	⊢ Ⅎ 𝑓 ( 𝜒 ∧ 𝑧 ∈ 𝐸 )
37	17 36	nfxfr	⊢ Ⅎ 𝑓 𝜃
38		nfv	⊢ Ⅎ 𝑓 𝑧 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 )
39	37 38	nfan	⊢ Ⅎ 𝑓 ( 𝜃 ∧ 𝑧 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) )
40	19 39	nfxfr	⊢ Ⅎ 𝑓 𝜁
41	40	nf5ri	⊢ ( 𝜁 → ∀ 𝑓 𝜁 )