Theorem bnj1447

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	bnj1447.1	⊢ 𝐵 = { 𝑑 ∣ ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
	bnj1447.2	⊢ 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
	bnj1447.3	⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) }
	bnj1447.4	⊢ ( 𝜏 ↔ ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
	bnj1447.5	⊢ 𝐷 = { 𝑥 ∈ 𝐴 ∣ ¬ ∃ 𝑓 𝜏 }
	bnj1447.6	⊢ ( 𝜓 ↔ ( 𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅ ) )
	bnj1447.7	⊢ ( 𝜒 ↔ ( 𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀ 𝑦 ∈ 𝐷 ¬ 𝑦 𝑅 𝑥 ) )
	bnj1447.8	⊢ ( 𝜏′ ↔ [ 𝑦 / 𝑥 ] 𝜏 )
	bnj1447.9	⊢ 𝐻 = { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) 𝜏′ }
	bnj1447.10	⊢ 𝑃 = ∪ 𝐻
	bnj1447.11	⊢ 𝑍 = ⟨ 𝑥 , ( 𝑃 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
	bnj1447.12	⊢ 𝑄 = ( 𝑃 ∪ { ⟨ 𝑥 , ( 𝐺 ‘ 𝑍 ) ⟩ } )
	bnj1447.13	⊢ 𝑊 = ⟨ 𝑧 , ( 𝑄 ↾ pred ( 𝑧 , 𝐴 , 𝑅 ) ) ⟩
Assertion	bnj1447	⊢ ( ( 𝑄 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑊 ) → ∀ 𝑦 ( 𝑄 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑊 ) )

Proof

Step	Hyp	Ref	Expression
1		bnj1447.1	⊢ 𝐵 = { 𝑑 ∣ ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
2		bnj1447.2	⊢ 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
3		bnj1447.3	⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) }
4		bnj1447.4	⊢ ( 𝜏 ↔ ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
5		bnj1447.5	⊢ 𝐷 = { 𝑥 ∈ 𝐴 ∣ ¬ ∃ 𝑓 𝜏 }
6		bnj1447.6	⊢ ( 𝜓 ↔ ( 𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅ ) )
7		bnj1447.7	⊢ ( 𝜒 ↔ ( 𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀ 𝑦 ∈ 𝐷 ¬ 𝑦 𝑅 𝑥 ) )
8		bnj1447.8	⊢ ( 𝜏′ ↔ [ 𝑦 / 𝑥 ] 𝜏 )
9		bnj1447.9	⊢ 𝐻 = { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) 𝜏′ }
10		bnj1447.10	⊢ 𝑃 = ∪ 𝐻
11		bnj1447.11	⊢ 𝑍 = ⟨ 𝑥 , ( 𝑃 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
12		bnj1447.12	⊢ 𝑄 = ( 𝑃 ∪ { ⟨ 𝑥 , ( 𝐺 ‘ 𝑍 ) ⟩ } )
13		bnj1447.13	⊢ 𝑊 = ⟨ 𝑧 , ( 𝑄 ↾ pred ( 𝑧 , 𝐴 , 𝑅 ) ) ⟩
14		nfre1	⊢ Ⅎ 𝑦 ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) 𝜏′
15	14	nfab	⊢ Ⅎ 𝑦 { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) 𝜏′ }
16	9 15	nfcxfr	⊢ Ⅎ 𝑦 𝐻
17	16	nfuni	⊢ Ⅎ 𝑦 ∪ 𝐻
18	10 17	nfcxfr	⊢ Ⅎ 𝑦 𝑃
19		nfcv	⊢ Ⅎ 𝑦 𝑥
20		nfcv	⊢ Ⅎ 𝑦 𝐺
21		nfcv	⊢ Ⅎ 𝑦 pred ( 𝑥 , 𝐴 , 𝑅 )
22	18 21	nfres	⊢ Ⅎ 𝑦 ( 𝑃 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) )
23	19 22	nfop	⊢ Ⅎ 𝑦 ⟨ 𝑥 , ( 𝑃 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
24	11 23	nfcxfr	⊢ Ⅎ 𝑦 𝑍
25	20 24	nffv	⊢ Ⅎ 𝑦 ( 𝐺 ‘ 𝑍 )
26	19 25	nfop	⊢ Ⅎ 𝑦 ⟨ 𝑥 , ( 𝐺 ‘ 𝑍 ) ⟩
27	26	nfsn	⊢ Ⅎ 𝑦 { ⟨ 𝑥 , ( 𝐺 ‘ 𝑍 ) ⟩ }
28	18 27	nfun	⊢ Ⅎ 𝑦 ( 𝑃 ∪ { ⟨ 𝑥 , ( 𝐺 ‘ 𝑍 ) ⟩ } )
29	12 28	nfcxfr	⊢ Ⅎ 𝑦 𝑄
30		nfcv	⊢ Ⅎ 𝑦 𝑧
31	29 30	nffv	⊢ Ⅎ 𝑦 ( 𝑄 ‘ 𝑧 )
32		nfcv	⊢ Ⅎ 𝑦 pred ( 𝑧 , 𝐴 , 𝑅 )
33	29 32	nfres	⊢ Ⅎ 𝑦 ( 𝑄 ↾ pred ( 𝑧 , 𝐴 , 𝑅 ) )
34	30 33	nfop	⊢ Ⅎ 𝑦 ⟨ 𝑧 , ( 𝑄 ↾ pred ( 𝑧 , 𝐴 , 𝑅 ) ) ⟩
35	13 34	nfcxfr	⊢ Ⅎ 𝑦 𝑊
36	20 35	nffv	⊢ Ⅎ 𝑦 ( 𝐺 ‘ 𝑊 )
37	31 36	nfeq	⊢ Ⅎ 𝑦 ( 𝑄 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑊 )
38	37	nf5ri	⊢ ( ( 𝑄 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑊 ) → ∀ 𝑦 ( 𝑄 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑊 ) )