Theorem caufval 25306

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 25287	. 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 5868	. . . . . 6 ⊢ (𝑑 = 𝐷 → dom 𝑑 = dom 𝐷)
3	2	dmeqd 5870	. . . . 5 ⊢ (𝑑 = 𝐷 → dom dom 𝑑 = dom dom 𝐷)
4		xmetf 24358	. . . . . . . 8 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
5	4	fdmd 6687	. . . . . . 7 ⊢ (𝐷 ∈ (∞Met‘𝑋) → dom 𝐷 = (𝑋 × 𝑋))
6	5	dmeqd 5870	. . . . . 6 ⊢ (𝐷 ∈ (∞Met‘𝑋) → dom dom 𝐷 = dom (𝑋 × 𝑋))
7		dmxpid 5895	. . . . . 6 ⊢ dom (𝑋 × 𝑋) = 𝑋
8	6, 7	eqtrdi 2803	. . . . 5 ⊢ (𝐷 ∈ (∞Met‘𝑋) → dom dom 𝐷 = 𝑋)
9	3, 8	sylan9eqr 2809	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → dom dom 𝑑 = 𝑋)
10	9	oveq1d 7396	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → (dom dom 𝑑 ↑pm ℂ) = (𝑋 ↑pm ℂ))
11		simpr 487	. . . . . . . 8 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → 𝑑 = 𝐷)
12	11	fveq2d 6856	. . . . . . 7 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → (ball‘𝑑) = (ball‘𝐷))
13	12	oveqd 7398	. . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → ((𝑓‘𝑘)(ball‘𝑑)𝑥) = ((𝑓‘𝑘)(ball‘𝐷)𝑥))
14	13	feq3d 6661	. . . . 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 3176	. . . 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 3175	. . 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 3422	. 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 6883	. . 3 ⊢ (∞Met‘𝑋) ⊆ ∪ ran ∞Met
19	18	sseli 3923	. 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 ∈ ∪ ran ∞Met)
20		ovex 7414	. . . 4 ⊢ (𝑋 ↑pm ℂ) ∈ V
21	20	rabex 5285	. . 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 6969	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 398   = wceq 1550   ∈ wcel 2132  ∀wral 3066  ∃wrex 3076  {crab 3404  Vcvv 3444  ∪ cuni 4855   × cxp 5634  dom cdm 5636  ran crn 5637   ↾ cres 5638  ⟶wf 6502  ‘cfv 6506  (class class class)co 7381   ↑_pm cpm 8793  ℂcc 11057  ℝ^*cxr 11201  ℤcz 12554  ℤ_≥cuz 12825  ℝ⁺crp 12979  ∞Metcxmet 21378  ballcbl 21380  Cauccau 25284

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-10 2165  ax-11 2181  ax-12 2202  ax-ext 2724  ax-sep 5236  ax-nul 5246  ax-pow 5312  ax-pr 5380  ax-un 7703  ax-cnex 11115  ax-resscn 11116

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-nf 1794  df-sb 2081  df-mo 2556  df-eu 2586  df-clab 2731  df-cleq 2744  df-clel 2827  df-nfc 2901  df-ne 2948  df-ral 3067  df-rex 3077  df-rab 3405  df-v 3446  df-sbc 3736  df-csb 3844  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-nul 4277  df-if 4471  df-pw 4547  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4856  df-br 5091  df-opab 5153  df-mpt 5172  df-id 5531  df-xp 5642  df-rel 5643  df-cnv 5644  df-co 5645  df-dm 5646  df-rn 5647  df-iota 6462  df-fun 6508  df-fn 6509  df-f 6510  df-fv 6514  df-ov 7384  df-oprab 7385  df-mpo 7386  df-map 8794  df-xr 11206  df-xmet 21386  df-cau 25287

This theorem is referenced by:  iscau  25307  equivcau  25331