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Theorem caufval 23292
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 ∈ (𝑋pm ℂ) ∣ ∀𝑥 ∈ ℝ+𝑘 ∈ ℤ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(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.
StepHypRef Expression
1 df-cau 23273 . . 3 Cau = (𝑑 ran ∞Met ↦ {𝑓 ∈ (dom dom 𝑑pm ℂ) ∣ ∀𝑥 ∈ ℝ+𝑘 ∈ ℤ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝑑)𝑥)})
21a1i 11 . 2 (𝐷 ∈ (∞Met‘𝑋) → Cau = (𝑑 ran ∞Met ↦ {𝑓 ∈ (dom dom 𝑑pm ℂ) ∣ ∀𝑥 ∈ ℝ+𝑘 ∈ ℤ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝑑)𝑥)}))
3 dmeq 5462 . . . . . 6 (𝑑 = 𝐷 → dom 𝑑 = dom 𝐷)
43dmeqd 5464 . . . . 5 (𝑑 = 𝐷 → dom dom 𝑑 = dom dom 𝐷)
5 xmetf 22354 . . . . . . . 8 (𝐷 ∈ (∞Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
6 fdm 6191 . . . . . . . 8 (𝐷:(𝑋 × 𝑋)⟶ℝ* → dom 𝐷 = (𝑋 × 𝑋))
75, 6syl 17 . . . . . . 7 (𝐷 ∈ (∞Met‘𝑋) → dom 𝐷 = (𝑋 × 𝑋))
87dmeqd 5464 . . . . . 6 (𝐷 ∈ (∞Met‘𝑋) → dom dom 𝐷 = dom (𝑋 × 𝑋))
9 dmxpid 5483 . . . . . 6 dom (𝑋 × 𝑋) = 𝑋
108, 9syl6eq 2821 . . . . 5 (𝐷 ∈ (∞Met‘𝑋) → dom dom 𝐷 = 𝑋)
114, 10sylan9eqr 2827 . . . 4 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → dom dom 𝑑 = 𝑋)
1211oveq1d 6808 . . 3 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → (dom dom 𝑑pm ℂ) = (𝑋pm ℂ))
13 simpr 471 . . . . . . . 8 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → 𝑑 = 𝐷)
1413fveq2d 6336 . . . . . . 7 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → (ball‘𝑑) = (ball‘𝐷))
1514oveqd 6810 . . . . . 6 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → ((𝑓𝑘)(ball‘𝑑)𝑥) = ((𝑓𝑘)(ball‘𝐷)𝑥))
1615feq3d 6172 . . . . 5 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → ((𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝑑)𝑥) ↔ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐷)𝑥)))
1716rexbidv 3200 . . . 4 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → (∃𝑘 ∈ ℤ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝑑)𝑥) ↔ ∃𝑘 ∈ ℤ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐷)𝑥)))
1817ralbidv 3135 . . 3 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → (∀𝑥 ∈ ℝ+𝑘 ∈ ℤ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝑑)𝑥) ↔ ∀𝑥 ∈ ℝ+𝑘 ∈ ℤ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐷)𝑥)))
1912, 18rabeqbidv 3345 . 2 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → {𝑓 ∈ (dom dom 𝑑pm ℂ) ∣ ∀𝑥 ∈ ℝ+𝑘 ∈ ℤ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝑑)𝑥)} = {𝑓 ∈ (𝑋pm ℂ) ∣ ∀𝑥 ∈ ℝ+𝑘 ∈ ℤ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐷)𝑥)})
20 fvssunirn 6358 . . 3 (∞Met‘𝑋) ⊆ ran ∞Met
2120sseli 3748 . 2 (𝐷 ∈ (∞Met‘𝑋) → 𝐷 ran ∞Met)
22 ovex 6823 . . . 4 (𝑋pm ℂ) ∈ V
2322rabex 4946 . . 3 {𝑓 ∈ (𝑋pm ℂ) ∣ ∀𝑥 ∈ ℝ+𝑘 ∈ ℤ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐷)𝑥)} ∈ V
2423a1i 11 . 2 (𝐷 ∈ (∞Met‘𝑋) → {𝑓 ∈ (𝑋pm ℂ) ∣ ∀𝑥 ∈ ℝ+𝑘 ∈ ℤ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐷)𝑥)} ∈ V)
252, 19, 21, 24fvmptd 6430 1 (𝐷 ∈ (∞Met‘𝑋) → (Cau‘𝐷) = {𝑓 ∈ (𝑋pm ℂ) ∣ ∀𝑥 ∈ ℝ+𝑘 ∈ ℤ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐷)𝑥)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  wral 3061  wrex 3062  {crab 3065  Vcvv 3351   cuni 4574  cmpt 4863   × cxp 5247  dom cdm 5249  ran crn 5250  cres 5251  wf 6027  cfv 6031  (class class class)co 6793  pm cpm 8010  cc 10136  *cxr 10275  cz 11579  cuz 11888  +crp 12035  ∞Metcxmt 19946  ballcbl 19948  Caucca 23270
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-map 8011  df-xr 10280  df-xmet 19954  df-cau 23273
This theorem is referenced by:  iscau  23293  equivcau  23317
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