Theorem caufval

Description: The set of Cauchy sequences on a metric space. (Contributed by NM, 8-Sep-2006) (Revised by Mario Carneiro, 5-Sep-2015)

	Ref	Expression
Assertion	caufval	⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ( Cau ‘ 𝐷 ) = { 𝑓 ∈ ( 𝑋 ↑pm ℂ ) ∣ ∀ 𝑥 ∈ ℝ+ ∃ 𝑘 ∈ ℤ ( 𝑓 ↾ ( ℤ≥ ‘ 𝑘 ) ) : ( ℤ≥ ‘ 𝑘 ) ⟶ ( ( 𝑓 ‘ 𝑘 ) ( ball ‘ 𝐷 ) 𝑥 ) } )

Proof

Step	Hyp	Ref	Expression
1		df-cau	⊢ Cau = ( 𝑑 ∈ ∪ ran ∞Met ↦ { 𝑓 ∈ ( dom dom 𝑑 ↑pm ℂ ) ∣ ∀ 𝑥 ∈ ℝ+ ∃ 𝑘 ∈ ℤ ( 𝑓 ↾ ( ℤ≥ ‘ 𝑘 ) ) : ( ℤ≥ ‘ 𝑘 ) ⟶ ( ( 𝑓 ‘ 𝑘 ) ( ball ‘ 𝑑 ) 𝑥 ) } )
2		dmeq	⊢ ( 𝑑 = 𝐷 → dom 𝑑 = dom 𝐷 )
3	2	dmeqd	⊢ ( 𝑑 = 𝐷 → dom dom 𝑑 = dom dom 𝐷 )
4		xmetf	⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → 𝐷 : ( 𝑋 × 𝑋 ) ⟶ ℝ* )
5	4	fdmd	⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → dom 𝐷 = ( 𝑋 × 𝑋 ) )
6	5	dmeqd	⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → dom dom 𝐷 = dom ( 𝑋 × 𝑋 ) )
7		dmxpid	⊢ dom ( 𝑋 × 𝑋 ) = 𝑋
8	6 7	eqtrdi	⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → dom dom 𝐷 = 𝑋 )
9	3 8	sylan9eqr	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑑 = 𝐷 ) → dom dom 𝑑 = 𝑋 )
10	9	oveq1d	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑑 = 𝐷 ) → ( dom dom 𝑑 ↑pm ℂ ) = ( 𝑋 ↑pm ℂ ) )
11		simpr	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑑 = 𝐷 ) → 𝑑 = 𝐷 )
12	11	fveq2d	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑑 = 𝐷 ) → ( ball ‘ 𝑑 ) = ( ball ‘ 𝐷 ) )
13	12	oveqd	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑑 = 𝐷 ) → ( ( 𝑓 ‘ 𝑘 ) ( ball ‘ 𝑑 ) 𝑥 ) = ( ( 𝑓 ‘ 𝑘 ) ( ball ‘ 𝐷 ) 𝑥 ) )
14	13	feq3d	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑑 = 𝐷 ) → ( ( 𝑓 ↾ ( ℤ≥ ‘ 𝑘 ) ) : ( ℤ≥ ‘ 𝑘 ) ⟶ ( ( 𝑓 ‘ 𝑘 ) ( ball ‘ 𝑑 ) 𝑥 ) ↔ ( 𝑓 ↾ ( ℤ≥ ‘ 𝑘 ) ) : ( ℤ≥ ‘ 𝑘 ) ⟶ ( ( 𝑓 ‘ 𝑘 ) ( ball ‘ 𝐷 ) 𝑥 ) ) )
15	14	rexbidv	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑑 = 𝐷 ) → ( ∃ 𝑘 ∈ ℤ ( 𝑓 ↾ ( ℤ≥ ‘ 𝑘 ) ) : ( ℤ≥ ‘ 𝑘 ) ⟶ ( ( 𝑓 ‘ 𝑘 ) ( ball ‘ 𝑑 ) 𝑥 ) ↔ ∃ 𝑘 ∈ ℤ ( 𝑓 ↾ ( ℤ≥ ‘ 𝑘 ) ) : ( ℤ≥ ‘ 𝑘 ) ⟶ ( ( 𝑓 ‘ 𝑘 ) ( ball ‘ 𝐷 ) 𝑥 ) ) )
16	15	ralbidv	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑑 = 𝐷 ) → ( ∀ 𝑥 ∈ ℝ+ ∃ 𝑘 ∈ ℤ ( 𝑓 ↾ ( ℤ≥ ‘ 𝑘 ) ) : ( ℤ≥ ‘ 𝑘 ) ⟶ ( ( 𝑓 ‘ 𝑘 ) ( ball ‘ 𝑑 ) 𝑥 ) ↔ ∀ 𝑥 ∈ ℝ+ ∃ 𝑘 ∈ ℤ ( 𝑓 ↾ ( ℤ≥ ‘ 𝑘 ) ) : ( ℤ≥ ‘ 𝑘 ) ⟶ ( ( 𝑓 ‘ 𝑘 ) ( ball ‘ 𝐷 ) 𝑥 ) ) )
17	10 16	rabeqbidv	⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑑 = 𝐷 ) → { 𝑓 ∈ ( dom dom 𝑑 ↑pm ℂ ) ∣ ∀ 𝑥 ∈ ℝ+ ∃ 𝑘 ∈ ℤ ( 𝑓 ↾ ( ℤ≥ ‘ 𝑘 ) ) : ( ℤ≥ ‘ 𝑘 ) ⟶ ( ( 𝑓 ‘ 𝑘 ) ( ball ‘ 𝑑 ) 𝑥 ) } = { 𝑓 ∈ ( 𝑋 ↑pm ℂ ) ∣ ∀ 𝑥 ∈ ℝ+ ∃ 𝑘 ∈ ℤ ( 𝑓 ↾ ( ℤ≥ ‘ 𝑘 ) ) : ( ℤ≥ ‘ 𝑘 ) ⟶ ( ( 𝑓 ‘ 𝑘 ) ( ball ‘ 𝐷 ) 𝑥 ) } )
18		fvssunirn	⊢ ( ∞Met ‘ 𝑋 ) ⊆ ∪ ran ∞Met
19	18	sseli	⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → 𝐷 ∈ ∪ ran ∞Met )
20		ovex	⊢ ( 𝑋 ↑pm ℂ ) ∈ V
21	20	rabex	⊢ { 𝑓 ∈ ( 𝑋 ↑pm ℂ ) ∣ ∀ 𝑥 ∈ ℝ+ ∃ 𝑘 ∈ ℤ ( 𝑓 ↾ ( ℤ≥ ‘ 𝑘 ) ) : ( ℤ≥ ‘ 𝑘 ) ⟶ ( ( 𝑓 ‘ 𝑘 ) ( ball ‘ 𝐷 ) 𝑥 ) } ∈ V
22	21	a1i	⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → { 𝑓 ∈ ( 𝑋 ↑pm ℂ ) ∣ ∀ 𝑥 ∈ ℝ+ ∃ 𝑘 ∈ ℤ ( 𝑓 ↾ ( ℤ≥ ‘ 𝑘 ) ) : ( ℤ≥ ‘ 𝑘 ) ⟶ ( ( 𝑓 ‘ 𝑘 ) ( ball ‘ 𝐷 ) 𝑥 ) } ∈ V )
23	1 17 19 22	fvmptd2	⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ( Cau ‘ 𝐷 ) = { 𝑓 ∈ ( 𝑋 ↑pm ℂ ) ∣ ∀ 𝑥 ∈ ℝ+ ∃ 𝑘 ∈ ℤ ( 𝑓 ↾ ( ℤ≥ ‘ 𝑘 ) ) : ( ℤ≥ ‘ 𝑘 ) ⟶ ( ( 𝑓 ‘ 𝑘 ) ( ball ‘ 𝐷 ) 𝑥 ) } )