Theorem bnj1520

Description: Technical lemma for bnj1500 . 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	bnj1520.1	⊢ 𝐵 = { 𝑑 ∣ ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
	bnj1520.2	⊢ 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
	bnj1520.3	⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) }
	bnj1520.4	⊢ 𝐹 = ∪ 𝐶
Assertion	bnj1520	⊢ ( ( 𝐹 ‘ 𝑥 ) = ( 𝐺 ‘ ⟨ 𝑥 , ( 𝐹 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩ ) → ∀ 𝑓 ( 𝐹 ‘ 𝑥 ) = ( 𝐺 ‘ ⟨ 𝑥 , ( 𝐹 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩ ) )

Proof

Step	Hyp	Ref	Expression
1		bnj1520.1	⊢ 𝐵 = { 𝑑 ∣ ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
2		bnj1520.2	⊢ 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
3		bnj1520.3	⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) }
4		bnj1520.4	⊢ 𝐹 = ∪ 𝐶
5	3	bnj1317	⊢ ( 𝑤 ∈ 𝐶 → ∀ 𝑓 𝑤 ∈ 𝐶 )
6	5	nfcii	⊢ Ⅎ 𝑓 𝐶
7	6	nfuni	⊢ Ⅎ 𝑓 ∪ 𝐶
8	4 7	nfcxfr	⊢ Ⅎ 𝑓 𝐹
9		nfcv	⊢ Ⅎ 𝑓 𝑥
10	8 9	nffv	⊢ Ⅎ 𝑓 ( 𝐹 ‘ 𝑥 )
11		nfcv	⊢ Ⅎ 𝑓 𝐺
12		nfcv	⊢ Ⅎ 𝑓 pred ( 𝑥 , 𝐴 , 𝑅 )
13	8 12	nfres	⊢ Ⅎ 𝑓 ( 𝐹 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) )
14	9 13	nfop	⊢ Ⅎ 𝑓 ⟨ 𝑥 , ( 𝐹 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
15	11 14	nffv	⊢ Ⅎ 𝑓 ( 𝐺 ‘ ⟨ 𝑥 , ( 𝐹 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩ )
16	10 15	nfeq	⊢ Ⅎ 𝑓 ( 𝐹 ‘ 𝑥 ) = ( 𝐺 ‘ ⟨ 𝑥 , ( 𝐹 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩ )
17	16	nf5ri	⊢ ( ( 𝐹 ‘ 𝑥 ) = ( 𝐺 ‘ ⟨ 𝑥 , ( 𝐹 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩ ) → ∀ 𝑓 ( 𝐹 ‘ 𝑥 ) = ( 𝐺 ‘ ⟨ 𝑥 , ( 𝐹 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩ ) )