Theorem bnj1446

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	bnj1446.1	⊢ 𝐵 = { 𝑑 ∣ ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
	bnj1446.2	⊢ 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
	bnj1446.3	⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) }
	bnj1446.4	⊢ ( 𝜏 ↔ ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
	bnj1446.5	⊢ 𝐷 = { 𝑥 ∈ 𝐴 ∣ ¬ ∃ 𝑓 𝜏 }
	bnj1446.6	⊢ ( 𝜓 ↔ ( 𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅ ) )
	bnj1446.7	⊢ ( 𝜒 ↔ ( 𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀ 𝑦 ∈ 𝐷 ¬ 𝑦 𝑅 𝑥 ) )
	bnj1446.8	⊢ ( 𝜏′ ↔ [ 𝑦 / 𝑥 ] 𝜏 )
	bnj1446.9	⊢ 𝐻 = { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) 𝜏′ }
	bnj1446.10	⊢ 𝑃 = ∪ 𝐻
	bnj1446.11	⊢ 𝑍 = ⟨ 𝑥 , ( 𝑃 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
	bnj1446.12	⊢ 𝑄 = ( 𝑃 ∪ { ⟨ 𝑥 , ( 𝐺 ‘ 𝑍 ) ⟩ } )
	bnj1446.13	⊢ 𝑊 = ⟨ 𝑧 , ( 𝑄 ↾ pred ( 𝑧 , 𝐴 , 𝑅 ) ) ⟩
Assertion	bnj1446	⊢ ( ( 𝑄 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑊 ) → ∀ 𝑑 ( 𝑄 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑊 ) )

Proof

Step	Hyp	Ref	Expression
1		bnj1446.1	⊢ 𝐵 = { 𝑑 ∣ ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
2		bnj1446.2	⊢ 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
3		bnj1446.3	⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) }
4		bnj1446.4	⊢ ( 𝜏 ↔ ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
5		bnj1446.5	⊢ 𝐷 = { 𝑥 ∈ 𝐴 ∣ ¬ ∃ 𝑓 𝜏 }
6		bnj1446.6	⊢ ( 𝜓 ↔ ( 𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅ ) )
7		bnj1446.7	⊢ ( 𝜒 ↔ ( 𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀ 𝑦 ∈ 𝐷 ¬ 𝑦 𝑅 𝑥 ) )
8		bnj1446.8	⊢ ( 𝜏′ ↔ [ 𝑦 / 𝑥 ] 𝜏 )
9		bnj1446.9	⊢ 𝐻 = { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) 𝜏′ }
10		bnj1446.10	⊢ 𝑃 = ∪ 𝐻
11		bnj1446.11	⊢ 𝑍 = ⟨ 𝑥 , ( 𝑃 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
12		bnj1446.12	⊢ 𝑄 = ( 𝑃 ∪ { ⟨ 𝑥 , ( 𝐺 ‘ 𝑍 ) ⟩ } )
13		bnj1446.13	⊢ 𝑊 = ⟨ 𝑧 , ( 𝑄 ↾ pred ( 𝑧 , 𝐴 , 𝑅 ) ) ⟩
14		nfcv	⊢ Ⅎ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 )
15		nfcv	⊢ Ⅎ 𝑑 𝑦
16		nfre1	⊢ Ⅎ 𝑑 ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) )
17	16	nfab	⊢ Ⅎ 𝑑 { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) }
18	3 17	nfcxfr	⊢ Ⅎ 𝑑 𝐶
19	18	nfcri	⊢ Ⅎ 𝑑 𝑓 ∈ 𝐶
20		nfv	⊢ Ⅎ 𝑑 dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) )
21	19 20	nfan	⊢ Ⅎ 𝑑 ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) )
22	4 21	nfxfr	⊢ Ⅎ 𝑑 𝜏
23	15 22	nfsbcw	⊢ Ⅎ 𝑑 [ 𝑦 / 𝑥 ] 𝜏
24	8 23	nfxfr	⊢ Ⅎ 𝑑 𝜏′
25	14 24	nfrex	⊢ Ⅎ 𝑑 ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) 𝜏′
26	25	nfab	⊢ Ⅎ 𝑑 { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) 𝜏′ }
27	9 26	nfcxfr	⊢ Ⅎ 𝑑 𝐻
28	27	nfuni	⊢ Ⅎ 𝑑 ∪ 𝐻
29	10 28	nfcxfr	⊢ Ⅎ 𝑑 𝑃
30		nfcv	⊢ Ⅎ 𝑑 𝑥
31		nfcv	⊢ Ⅎ 𝑑 𝐺
32	29 14	nfres	⊢ Ⅎ 𝑑 ( 𝑃 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) )
33	30 32	nfop	⊢ Ⅎ 𝑑 ⟨ 𝑥 , ( 𝑃 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
34	11 33	nfcxfr	⊢ Ⅎ 𝑑 𝑍
35	31 34	nffv	⊢ Ⅎ 𝑑 ( 𝐺 ‘ 𝑍 )
36	30 35	nfop	⊢ Ⅎ 𝑑 ⟨ 𝑥 , ( 𝐺 ‘ 𝑍 ) ⟩
37	36	nfsn	⊢ Ⅎ 𝑑 { ⟨ 𝑥 , ( 𝐺 ‘ 𝑍 ) ⟩ }
38	29 37	nfun	⊢ Ⅎ 𝑑 ( 𝑃 ∪ { ⟨ 𝑥 , ( 𝐺 ‘ 𝑍 ) ⟩ } )
39	12 38	nfcxfr	⊢ Ⅎ 𝑑 𝑄
40		nfcv	⊢ Ⅎ 𝑑 𝑧
41	39 40	nffv	⊢ Ⅎ 𝑑 ( 𝑄 ‘ 𝑧 )
42		nfcv	⊢ Ⅎ 𝑑 pred ( 𝑧 , 𝐴 , 𝑅 )
43	39 42	nfres	⊢ Ⅎ 𝑑 ( 𝑄 ↾ pred ( 𝑧 , 𝐴 , 𝑅 ) )
44	40 43	nfop	⊢ Ⅎ 𝑑 ⟨ 𝑧 , ( 𝑄 ↾ pred ( 𝑧 , 𝐴 , 𝑅 ) ) ⟩
45	13 44	nfcxfr	⊢ Ⅎ 𝑑 𝑊
46	31 45	nffv	⊢ Ⅎ 𝑑 ( 𝐺 ‘ 𝑊 )
47	41 46	nfeq	⊢ Ⅎ 𝑑 ( 𝑄 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑊 )
48	47	nf5ri	⊢ ( ( 𝑄 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑊 ) → ∀ 𝑑 ( 𝑄 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑊 ) )