Theorem bnj1525

Description: Technical lemma for bnj1522 . 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	bnj1525.1	⊢ 𝐵 = { 𝑑 ∣ ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
	bnj1525.2	⊢ 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
	bnj1525.3	⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) }
	bnj1525.4	⊢ 𝐹 = ∪ 𝐶
	bnj1525.5	⊢ ( 𝜑 ↔ ( 𝑅 FrSe 𝐴 ∧ 𝐻 Fn 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐻 ‘ 𝑥 ) = ( 𝐺 ‘ ⟨ 𝑥 , ( 𝐻 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩ ) ) )
	bnj1525.6	⊢ ( 𝜓 ↔ ( 𝜑 ∧ 𝐹 ≠ 𝐻 ) )
Assertion	bnj1525	⊢ ( 𝜓 → ∀ 𝑥 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		bnj1525.1	⊢ 𝐵 = { 𝑑 ∣ ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
2		bnj1525.2	⊢ 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
3		bnj1525.3	⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) }
4		bnj1525.4	⊢ 𝐹 = ∪ 𝐶
5		bnj1525.5	⊢ ( 𝜑 ↔ ( 𝑅 FrSe 𝐴 ∧ 𝐻 Fn 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐻 ‘ 𝑥 ) = ( 𝐺 ‘ ⟨ 𝑥 , ( 𝐻 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩ ) ) )
6		bnj1525.6	⊢ ( 𝜓 ↔ ( 𝜑 ∧ 𝐹 ≠ 𝐻 ) )
7		nfv	⊢ Ⅎ 𝑥 𝑅 FrSe 𝐴
8		nfv	⊢ Ⅎ 𝑥 𝐻 Fn 𝐴
9		nfra1	⊢ Ⅎ 𝑥 ∀ 𝑥 ∈ 𝐴 ( 𝐻 ‘ 𝑥 ) = ( 𝐺 ‘ ⟨ 𝑥 , ( 𝐻 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩ )
10	7 8 9	nf3an	⊢ Ⅎ 𝑥 ( 𝑅 FrSe 𝐴 ∧ 𝐻 Fn 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐻 ‘ 𝑥 ) = ( 𝐺 ‘ ⟨ 𝑥 , ( 𝐻 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩ ) )
11	5 10	nfxfr	⊢ Ⅎ 𝑥 𝜑
12	1	bnj1309	⊢ ( 𝑤 ∈ 𝐵 → ∀ 𝑥 𝑤 ∈ 𝐵 )
13	3 12	bnj1307	⊢ ( 𝑤 ∈ 𝐶 → ∀ 𝑥 𝑤 ∈ 𝐶 )
14	13	nfcii	⊢ Ⅎ 𝑥 𝐶
15	14	nfuni	⊢ Ⅎ 𝑥 ∪ 𝐶
16	4 15	nfcxfr	⊢ Ⅎ 𝑥 𝐹
17		nfcv	⊢ Ⅎ 𝑥 𝐻
18	16 17	nfne	⊢ Ⅎ 𝑥 𝐹 ≠ 𝐻
19	11 18	nfan	⊢ Ⅎ 𝑥 ( 𝜑 ∧ 𝐹 ≠ 𝐻 )
20	6 19	nfxfr	⊢ Ⅎ 𝑥 𝜓
21	20	nf5ri	⊢ ( 𝜓 → ∀ 𝑥 𝜓 )