Theorem bnj1448

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	bnj1448.1	⊢ 𝐵 = { 𝑑 ∣ ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
	bnj1448.2	⊢ 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
	bnj1448.3	⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) }
	bnj1448.4	⊢ ( 𝜏 ↔ ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
	bnj1448.5	⊢ 𝐷 = { 𝑥 ∈ 𝐴 ∣ ¬ ∃ 𝑓 𝜏 }
	bnj1448.6	⊢ ( 𝜓 ↔ ( 𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅ ) )
	bnj1448.7	⊢ ( 𝜒 ↔ ( 𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀ 𝑦 ∈ 𝐷 ¬ 𝑦 𝑅 𝑥 ) )
	bnj1448.8	⊢ ( 𝜏′ ↔ [ 𝑦 / 𝑥 ] 𝜏 )
	bnj1448.9	⊢ 𝐻 = { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) 𝜏′ }
	bnj1448.10	⊢ 𝑃 = ∪ 𝐻
	bnj1448.11	⊢ 𝑍 = ⟨ 𝑥 , ( 𝑃 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
	bnj1448.12	⊢ 𝑄 = ( 𝑃 ∪ { ⟨ 𝑥 , ( 𝐺 ‘ 𝑍 ) ⟩ } )
	bnj1448.13	⊢ 𝑊 = ⟨ 𝑧 , ( 𝑄 ↾ pred ( 𝑧 , 𝐴 , 𝑅 ) ) ⟩
Assertion	bnj1448	⊢ ( ( 𝑄 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑊 ) → ∀ 𝑓 ( 𝑄 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑊 ) )

Proof

Step	Hyp	Ref	Expression
1		bnj1448.1	⊢ 𝐵 = { 𝑑 ∣ ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
2		bnj1448.2	⊢ 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
3		bnj1448.3	⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) }
4		bnj1448.4	⊢ ( 𝜏 ↔ ( 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ( { 𝑥 } ∪ trCl ( 𝑥 , 𝐴 , 𝑅 ) ) ) )
5		bnj1448.5	⊢ 𝐷 = { 𝑥 ∈ 𝐴 ∣ ¬ ∃ 𝑓 𝜏 }
6		bnj1448.6	⊢ ( 𝜓 ↔ ( 𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅ ) )
7		bnj1448.7	⊢ ( 𝜒 ↔ ( 𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀ 𝑦 ∈ 𝐷 ¬ 𝑦 𝑅 𝑥 ) )
8		bnj1448.8	⊢ ( 𝜏′ ↔ [ 𝑦 / 𝑥 ] 𝜏 )
9		bnj1448.9	⊢ 𝐻 = { 𝑓 ∣ ∃ 𝑦 ∈ pred ( 𝑥 , 𝐴 , 𝑅 ) 𝜏′ }
10		bnj1448.10	⊢ 𝑃 = ∪ 𝐻
11		bnj1448.11	⊢ 𝑍 = ⟨ 𝑥 , ( 𝑃 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
12		bnj1448.12	⊢ 𝑄 = ( 𝑃 ∪ { ⟨ 𝑥 , ( 𝐺 ‘ 𝑍 ) ⟩ } )
13		bnj1448.13	⊢ 𝑊 = ⟨ 𝑧 , ( 𝑄 ↾ pred ( 𝑧 , 𝐴 , 𝑅 ) ) ⟩
14	9	bnj1317	⊢ ( 𝑤 ∈ 𝐻 → ∀ 𝑓 𝑤 ∈ 𝐻 )
15	14	nfcii	⊢ Ⅎ 𝑓 𝐻
16	15	nfuni	⊢ Ⅎ 𝑓 ∪ 𝐻
17	10 16	nfcxfr	⊢ Ⅎ 𝑓 𝑃
18		nfcv	⊢ Ⅎ 𝑓 𝑥
19		nfcv	⊢ Ⅎ 𝑓 𝐺
20		nfcv	⊢ Ⅎ 𝑓 pred ( 𝑥 , 𝐴 , 𝑅 )
21	17 20	nfres	⊢ Ⅎ 𝑓 ( 𝑃 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) )
22	18 21	nfop	⊢ Ⅎ 𝑓 ⟨ 𝑥 , ( 𝑃 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
23	11 22	nfcxfr	⊢ Ⅎ 𝑓 𝑍
24	19 23	nffv	⊢ Ⅎ 𝑓 ( 𝐺 ‘ 𝑍 )
25	18 24	nfop	⊢ Ⅎ 𝑓 ⟨ 𝑥 , ( 𝐺 ‘ 𝑍 ) ⟩
26	25	nfsn	⊢ Ⅎ 𝑓 { ⟨ 𝑥 , ( 𝐺 ‘ 𝑍 ) ⟩ }
27	17 26	nfun	⊢ Ⅎ 𝑓 ( 𝑃 ∪ { ⟨ 𝑥 , ( 𝐺 ‘ 𝑍 ) ⟩ } )
28	12 27	nfcxfr	⊢ Ⅎ 𝑓 𝑄
29		nfcv	⊢ Ⅎ 𝑓 𝑧
30	28 29	nffv	⊢ Ⅎ 𝑓 ( 𝑄 ‘ 𝑧 )
31		nfcv	⊢ Ⅎ 𝑓 pred ( 𝑧 , 𝐴 , 𝑅 )
32	28 31	nfres	⊢ Ⅎ 𝑓 ( 𝑄 ↾ pred ( 𝑧 , 𝐴 , 𝑅 ) )
33	29 32	nfop	⊢ Ⅎ 𝑓 ⟨ 𝑧 , ( 𝑄 ↾ pred ( 𝑧 , 𝐴 , 𝑅 ) ) ⟩
34	13 33	nfcxfr	⊢ Ⅎ 𝑓 𝑊
35	19 34	nffv	⊢ Ⅎ 𝑓 ( 𝐺 ‘ 𝑊 )
36	30 35	nfeq	⊢ Ⅎ 𝑓 ( 𝑄 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑊 )
37	36	nf5ri	⊢ ( ( 𝑄 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑊 ) → ∀ 𝑓 ( 𝑄 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑊 ) )