Theorem caufval 25231

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

Assertion

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

Distinct variable groups:   𝑓,𝑘,𝑥,𝐷   𝑓,𝑋,𝑘,𝑥

Proof of Theorem caufval

Dummy variable 𝑑 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-cau 25212	. 2 ⊢ 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 5852	. . . . . 6 ⊢ (𝑑 = 𝐷 → dom 𝑑 = dom 𝐷)
3	2	dmeqd 5854	. . . . 5 ⊢ (𝑑 = 𝐷 → dom dom 𝑑 = dom dom 𝐷)
4		xmetf 24273	. . . . . . . 8 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
5	4	fdmd 6672	. . . . . . 7 ⊢ (𝐷 ∈ (∞Met‘𝑋) → dom 𝐷 = (𝑋 × 𝑋))
6	5	dmeqd 5854	. . . . . 6 ⊢ (𝐷 ∈ (∞Met‘𝑋) → dom dom 𝐷 = dom (𝑋 × 𝑋))
7		dmxpid 5879	. . . . . 6 ⊢ dom (𝑋 × 𝑋) = 𝑋
8	6, 7	eqtrdi 2787	. . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑋) → dom dom 𝐷 = 𝑋)
9	3, 8	sylan9eqr 2793	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → dom dom 𝑑 = 𝑋)
10	9	oveq1d 7373	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → (dom dom 𝑑 ↑pm ℂ) = (𝑋 ↑pm ℂ))
11		simpr 484	. . . . . . . 8 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → 𝑑 = 𝐷)
12	11	fveq2d 6838	. . . . . . 7 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → (ball‘𝑑) = (ball‘𝐷))
13	12	oveqd 7375	. . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → ((𝑓‘𝑘)(ball‘𝑑)𝑥) = ((𝑓‘𝑘)(ball‘𝐷)𝑥))
14	13	feq3d 6647	. . . . 5 ⊢ ((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 3160	. . . 4 ⊢ ((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 3159	. . 3 ⊢ ((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 3417	. 2 ⊢ ((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 6865	. . 3 ⊢ (∞Met‘𝑋) ⊆ ∪ ran ∞Met
19	18	sseli 3929	. 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 ∈ ∪ ran ∞Met)
20		ovex 7391	. . . 4 ⊢ (𝑋 ↑pm ℂ) ∈ V
21	20	rabex 5284	. . 3 ⊢ {𝑓 ∈ (𝑋 ↑pm ℂ) ∣ ∀𝑥 ∈ ℝ+ ∃𝑘 ∈ ℤ (𝑓 ↾ (ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶((𝑓‘𝑘)(ball‘𝐷)𝑥)} ∈ V
22	21	a1i 11	. 2 ⊢ (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 6949	1 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (Cau‘𝐷) = {𝑓 ∈ (𝑋 ↑pm ℂ) ∣ ∀𝑥 ∈ ℝ+ ∃𝑘 ∈ ℤ (𝑓 ↾ (ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶((𝑓‘𝑘)(ball‘𝐷)𝑥)})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3051  ∃wrex 3060  {crab 3399  Vcvv 3440  ∪ cuni 4863   × cxp 5622  dom cdm 5624  ran crn 5625   ↾ cres 5626  ⟶wf 6488  ‘cfv 6492  (class class class)co 7358   ↑_pm cpm 8764  ℂcc 11024  ℝ^*cxr 11165  ℤcz 12488  ℤ_≥cuz 12751  ℝ⁺crp 12905  ∞Metcxmet 21294  ballcbl 21296  Cauccau 25209

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-map 8765  df-xr 11170  df-xmet 21302  df-cau 25212

This theorem is referenced by:  iscau  25232  equivcau  25256