Theorem bnj1518

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	bnj1518.1	⊢ 𝐵 = { 𝑑 ∣ ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
	bnj1518.2	⊢ 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
	bnj1518.3	⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) }
	bnj1518.4	⊢ 𝐹 = ∪ 𝐶
	bnj1518.5	⊢ ( 𝜑 ↔ ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) )
	bnj1518.6	⊢ ( 𝜓 ↔ ( 𝜑 ∧ 𝑓 ∈ 𝐶 ∧ 𝑥 ∈ dom 𝑓 ) )
Assertion	bnj1518	⊢ ( 𝜓 → ∀ 𝑑 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		bnj1518.1	⊢ 𝐵 = { 𝑑 ∣ ( 𝑑 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑑 pred ( 𝑥 , 𝐴 , 𝑅 ) ⊆ 𝑑 ) }
2		bnj1518.2	⊢ 𝑌 = ⟨ 𝑥 , ( 𝑓 ↾ pred ( 𝑥 , 𝐴 , 𝑅 ) ) ⟩
3		bnj1518.3	⊢ 𝐶 = { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) }
4		bnj1518.4	⊢ 𝐹 = ∪ 𝐶
5		bnj1518.5	⊢ ( 𝜑 ↔ ( 𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ) )
6		bnj1518.6	⊢ ( 𝜓 ↔ ( 𝜑 ∧ 𝑓 ∈ 𝐶 ∧ 𝑥 ∈ dom 𝑓 ) )
7		nfv	⊢ Ⅎ 𝑑 𝜑
8		nfre1	⊢ Ⅎ 𝑑 ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) )
9	8	nfab	⊢ Ⅎ 𝑑 { 𝑓 ∣ ∃ 𝑑 ∈ 𝐵 ( 𝑓 Fn 𝑑 ∧ ∀ 𝑥 ∈ 𝑑 ( 𝑓 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑌 ) ) }
10	3 9	nfcxfr	⊢ Ⅎ 𝑑 𝐶
11	10	nfcri	⊢ Ⅎ 𝑑 𝑓 ∈ 𝐶
12		nfv	⊢ Ⅎ 𝑑 𝑥 ∈ dom 𝑓
13	7 11 12	nf3an	⊢ Ⅎ 𝑑 ( 𝜑 ∧ 𝑓 ∈ 𝐶 ∧ 𝑥 ∈ dom 𝑓 )
14	6 13	nfxfr	⊢ Ⅎ 𝑑 𝜓
15	14	nf5ri	⊢ ( 𝜓 → ∀ 𝑑 𝜓 )