Theorem bnj1467

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	bnj1467.1	⊢ 𝐵 = { 𝑑 ∣ ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
	bnj1467.2	⊢ 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
	bnj1467.3	⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) }
	bnj1467.4	⊢ ( 𝜏 ↔ ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
	bnj1467.5	⊢ 𝐷 = { 𝑥 ∈ 𝐴 ∣ ¬ ∃ 𝑓 𝜏 }
	bnj1467.6	⊢ ( 𝜓 ↔ ( 𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅ ) )
	bnj1467.7	⊢ ( 𝜒 ↔ ( 𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀ 𝑦 ∈ 𝐷 ¬ 𝑦 𝑅 𝑥 ) )
	bnj1467.8	⊢ ( 𝜏′ ↔ [ 𝑦 / 𝑥 ] 𝜏 )
	bnj1467.9	⊢ 𝐻 = { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) 𝜏′ }
	bnj1467.10	⊢ 𝑃 = ∪ 𝐻
	bnj1467.11	⊢ 𝑍 = ⟨ 𝑥 , ( 𝑃 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
	bnj1467.12	⊢ 𝑄 = ( 𝑃 ∪ { ⟨ 𝑥 , ( 𝐺 ‘ 𝑍 ) ⟩ } )
Assertion	bnj1467	⊢ ( 𝑤 ∈ 𝑄 → ∀ 𝑑 𝑤 ∈ 𝑄 )

Proof

Step	Hyp	Ref	Expression
1		bnj1467.1	⊢ 𝐵 = { 𝑑 ∣ ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
2		bnj1467.2	⊢ 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
3		bnj1467.3	⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) }
4		bnj1467.4	⊢ ( 𝜏 ↔ ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
5		bnj1467.5	⊢ 𝐷 = { 𝑥 ∈ 𝐴 ∣ ¬ ∃ 𝑓 𝜏 }
6		bnj1467.6	⊢ ( 𝜓 ↔ ( 𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅ ) )
7		bnj1467.7	⊢ ( 𝜒 ↔ ( 𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀ 𝑦 ∈ 𝐷 ¬ 𝑦 𝑅 𝑥 ) )
8		bnj1467.8	⊢ ( 𝜏′ ↔ [ 𝑦 / 𝑥 ] 𝜏 )
9		bnj1467.9	⊢ 𝐻 = { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) 𝜏′ }
10		bnj1467.10	⊢ 𝑃 = ∪ 𝐻
11		bnj1467.11	⊢ 𝑍 = ⟨ 𝑥 , ( 𝑃 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
12		bnj1467.12	⊢ 𝑄 = ( 𝑃 ∪ { ⟨ 𝑥 , ( 𝐺 ‘ 𝑍 ) ⟩ } )
13		nfcv	⊢ Ⅎ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 )
14		nfcv	⊢ Ⅎ 𝑑 𝑦
15		nfre1	⊢ Ⅎ 𝑑 ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) )
16	15	nfab	⊢ Ⅎ 𝑑 { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) }
17	3 16	nfcxfr	⊢ Ⅎ 𝑑 𝐶
18	17	nfcri	⊢ Ⅎ 𝑑 𝑓 ∈ 𝐶
19		nfv	⊢ Ⅎ 𝑑 dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) )
20	18 19	nfan	⊢ Ⅎ 𝑑 ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) )
21	4 20	nfxfr	⊢ Ⅎ 𝑑 𝜏
22	14 21	nfsbcw	⊢ Ⅎ 𝑑 [ 𝑦 / 𝑥 ] 𝜏
23	8 22	nfxfr	⊢ Ⅎ 𝑑 𝜏′
24	13 23	nfrexw	⊢ Ⅎ 𝑑 ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) 𝜏′
25	24	nfab	⊢ Ⅎ 𝑑 { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) 𝜏′ }
26	9 25	nfcxfr	⊢ Ⅎ 𝑑 𝐻
27	26	nfuni	⊢ Ⅎ 𝑑 ∪ 𝐻
28	10 27	nfcxfr	⊢ Ⅎ 𝑑 𝑃
29		nfcv	⊢ Ⅎ 𝑑 𝑥
30		nfcv	⊢ Ⅎ 𝑑 𝐺
31	28 13	nfres	⊢ Ⅎ 𝑑 ( 𝑃 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) )
32	29 31	nfop	⊢ Ⅎ 𝑑 ⟨ 𝑥 , ( 𝑃 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
33	11 32	nfcxfr	⊢ Ⅎ 𝑑 𝑍
34	30 33	nffv	⊢ Ⅎ 𝑑 ( 𝐺 ‘ 𝑍 )
35	29 34	nfop	⊢ Ⅎ 𝑑 ⟨ 𝑥 , ( 𝐺 ‘ 𝑍 ) ⟩
36	35	nfsn	⊢ Ⅎ 𝑑 { ⟨ 𝑥 , ( 𝐺 ‘ 𝑍 ) ⟩ }
37	28 36	nfun	⊢ Ⅎ 𝑑 ( 𝑃 ∪ { ⟨ 𝑥 , ( 𝐺 ‘ 𝑍 ) ⟩ } )
38	12 37	nfcxfr	⊢ Ⅎ 𝑑 𝑄
39	38	nfcrii	⊢ ( 𝑤 ∈ 𝑄 → ∀ 𝑑 𝑤 ∈ 𝑄 )