bnj1444

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	bnj1444.1	⊢ 𝐵 = { 𝑑 ∣ ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
	bnj1444.2	⊢ 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
	bnj1444.3	⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) }
	bnj1444.4	⊢ ( 𝜏 ↔ ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
	bnj1444.5	⊢ 𝐷 = { 𝑥 ∈ 𝐴 ∣ ¬ ∃ 𝑓 𝜏 }
	bnj1444.6	⊢ ( 𝜓 ↔ ( 𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅ ) )
	bnj1444.7	⊢ ( 𝜒 ↔ ( 𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀ 𝑦 ∈ 𝐷 ¬ 𝑦 𝑅 𝑥 ) )
	bnj1444.8	⊢ ( 𝜏′ ↔ [ 𝑦 / 𝑥 ] 𝜏 )
	bnj1444.9	⊢ 𝐻 = { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) 𝜏′ }
	bnj1444.10	⊢ 𝑃 = ∪ 𝐻
	bnj1444.11	⊢ 𝑍 = ⟨ 𝑥 , ( 𝑃 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
	bnj1444.12	⊢ 𝑄 = ( 𝑃 ∪ { ⟨ 𝑥 , ( 𝐺 ‘ 𝑍 ) ⟩ } )
	bnj1444.13	⊢ 𝑊 = ⟨ 𝑧 , ( 𝑄 ↾ pred ( 𝑧 , 𝐴 , 𝑅 ) ) ⟩
	bnj1444.14	⊢ 𝐸 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) )
	bnj1444.15	⊢ ( 𝜒 → 𝑃 Fn trCl ( 𝑥 , 𝐴 , 𝑅 ) )
	bnj1444.16	⊢ ( 𝜒 → 𝑄 Fn ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) )
	bnj1444.17	⊢ ( 𝜃 ↔ ( 𝜒 ∧ 𝑧 ∈ 𝐸 ) )
	bnj1444.18	⊢ ( 𝜂 ↔ ( 𝜃 ∧ 𝑧 ∈ { 𝑥 } ) )
	bnj1444.19	⊢ ( 𝜁 ↔ ( 𝜃 ∧ 𝑧 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) )
	bnj1444.20	⊢ ( 𝜌 ↔ ( 𝜁 ∧ 𝑓 ∈ 𝐻 ∧ 𝑧 ∈ dom 𝑓 ) )
Assertion	bnj1444	⊢ ( 𝜌 → ∀ 𝑦 𝜌 )

Proof

Step	Hyp	Ref	Expression
1		bnj1444.1	⊢ 𝐵 = { 𝑑 ∣ ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
2		bnj1444.2	⊢ 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
3		bnj1444.3	⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) }
4		bnj1444.4	⊢ ( 𝜏 ↔ ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
5		bnj1444.5	⊢ 𝐷 = { 𝑥 ∈ 𝐴 ∣ ¬ ∃ 𝑓 𝜏 }
6		bnj1444.6	⊢ ( 𝜓 ↔ ( 𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅ ) )
7		bnj1444.7	⊢ ( 𝜒 ↔ ( 𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀ 𝑦 ∈ 𝐷 ¬ 𝑦 𝑅 𝑥 ) )
8		bnj1444.8	⊢ ( 𝜏′ ↔ [ 𝑦 / 𝑥 ] 𝜏 )
9		bnj1444.9	⊢ 𝐻 = { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) 𝜏′ }
10		bnj1444.10	⊢ 𝑃 = ∪ 𝐻
11		bnj1444.11	⊢ 𝑍 = ⟨ 𝑥 , ( 𝑃 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
12		bnj1444.12	⊢ 𝑄 = ( 𝑃 ∪ { ⟨ 𝑥 , ( 𝐺 ‘ 𝑍 ) ⟩ } )
13		bnj1444.13	⊢ 𝑊 = ⟨ 𝑧 , ( 𝑄 ↾ pred ( 𝑧 , 𝐴 , 𝑅 ) ) ⟩
14		bnj1444.14	⊢ 𝐸 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) )
15		bnj1444.15	⊢ ( 𝜒 → 𝑃 Fn trCl ( 𝑥 , 𝐴 , 𝑅 ) )
16		bnj1444.16	⊢ ( 𝜒 → 𝑄 Fn ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) )
17		bnj1444.17	⊢ ( 𝜃 ↔ ( 𝜒 ∧ 𝑧 ∈ 𝐸 ) )
18		bnj1444.18	⊢ ( 𝜂 ↔ ( 𝜃 ∧ 𝑧 ∈ { 𝑥 } ) )
19		bnj1444.19	⊢ ( 𝜁 ↔ ( 𝜃 ∧ 𝑧 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) )
20		bnj1444.20	⊢ ( 𝜌 ↔ ( 𝜁 ∧ 𝑓 ∈ 𝐻 ∧ 𝑧 ∈ dom 𝑓 ) )
21		nfv	⊢ Ⅎ 𝑦 𝜓
22		nfv	⊢ Ⅎ 𝑦 𝑥 ∈ 𝐷
23		nfra1	⊢ Ⅎ 𝑦 ∀ 𝑦 ∈ 𝐷 ¬ 𝑦 𝑅 𝑥
24	21 22 23	nf3an	⊢ Ⅎ 𝑦 ( 𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀ 𝑦 ∈ 𝐷 ¬ 𝑦 𝑅 𝑥 )
25	7 24	nfxfr	⊢ Ⅎ 𝑦 𝜒
26		nfv	⊢ Ⅎ 𝑦 𝑧 ∈ 𝐸
27	25 26	nfan	⊢ Ⅎ 𝑦 ( 𝜒 ∧ 𝑧 ∈ 𝐸 )
28	17 27	nfxfr	⊢ Ⅎ 𝑦 𝜃
29		nfv	⊢ Ⅎ 𝑦 𝑧 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 )
30	28 29	nfan	⊢ Ⅎ 𝑦 ( 𝜃 ∧ 𝑧 ∈ trCl ( 𝑥 , 𝐴 , 𝑅 ) )
31	19 30	nfxfr	⊢ Ⅎ 𝑦 𝜁
32		nfre1	⊢ Ⅎ 𝑦 ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) 𝜏′
33	32	nfab	⊢ Ⅎ 𝑦 { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) 𝜏′ }
34	9 33	nfcxfr	⊢ Ⅎ 𝑦 𝐻
35	34	nfcri	⊢ Ⅎ 𝑦 𝑓 ∈ 𝐻
36		nfv	⊢ Ⅎ 𝑦 𝑧 ∈ dom 𝑓
37	31 35 36	nf3an	⊢ Ⅎ 𝑦 ( 𝜁 ∧ 𝑓 ∈ 𝐻 ∧ 𝑧 ∈ dom 𝑓 )
38	20 37	nfxfr	⊢ Ⅎ 𝑦 𝜌
39	38	nf5ri	⊢ ( 𝜌 → ∀ 𝑦 𝜌 )

Metamath Proof Explorer

Theorem bnj1444

Proof