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Theorem iscmet 22890
 Description: The property "𝐷 is a complete metric." meaning all Cauchy filters converge to a point in the space. (Contributed by Mario Carneiro, 1-May-2014.) (Revised by Mario Carneiro, 13-Oct-2015.)
Hypothesis
Ref Expression
iscmet.1 𝐽 = (MetOpen‘𝐷)
Assertion
Ref Expression
iscmet (𝐷 ∈ (CMet‘𝑋) ↔ (𝐷 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑓) ≠ ∅))
Distinct variable groups:   𝐷,𝑓   𝑓,𝐽   𝑓,𝑋

Proof of Theorem iscmet
Dummy variables 𝑑 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvex 6131 . 2 (𝐷 ∈ (CMet‘𝑋) → 𝑋 ∈ V)
2 elfvex 6131 . . 3 (𝐷 ∈ (Met‘𝑋) → 𝑋 ∈ V)
32adantr 480 . 2 ((𝐷 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑓) ≠ ∅) → 𝑋 ∈ V)
4 fveq2 6103 . . . . . 6 (𝑥 = 𝑋 → (Met‘𝑥) = (Met‘𝑋))
5 rabeq 3166 . . . . . 6 ((Met‘𝑥) = (Met‘𝑋) → {𝑑 ∈ (Met‘𝑥) ∣ ∀𝑓 ∈ (CauFil‘𝑑)((MetOpen‘𝑑) fLim 𝑓) ≠ ∅} = {𝑑 ∈ (Met‘𝑋) ∣ ∀𝑓 ∈ (CauFil‘𝑑)((MetOpen‘𝑑) fLim 𝑓) ≠ ∅})
64, 5syl 17 . . . . 5 (𝑥 = 𝑋 → {𝑑 ∈ (Met‘𝑥) ∣ ∀𝑓 ∈ (CauFil‘𝑑)((MetOpen‘𝑑) fLim 𝑓) ≠ ∅} = {𝑑 ∈ (Met‘𝑋) ∣ ∀𝑓 ∈ (CauFil‘𝑑)((MetOpen‘𝑑) fLim 𝑓) ≠ ∅})
7 df-cmet 22863 . . . . 5 CMet = (𝑥 ∈ V ↦ {𝑑 ∈ (Met‘𝑥) ∣ ∀𝑓 ∈ (CauFil‘𝑑)((MetOpen‘𝑑) fLim 𝑓) ≠ ∅})
8 fvex 6113 . . . . . 6 (Met‘𝑋) ∈ V
98rabex 4740 . . . . 5 {𝑑 ∈ (Met‘𝑋) ∣ ∀𝑓 ∈ (CauFil‘𝑑)((MetOpen‘𝑑) fLim 𝑓) ≠ ∅} ∈ V
106, 7, 9fvmpt 6191 . . . 4 (𝑋 ∈ V → (CMet‘𝑋) = {𝑑 ∈ (Met‘𝑋) ∣ ∀𝑓 ∈ (CauFil‘𝑑)((MetOpen‘𝑑) fLim 𝑓) ≠ ∅})
1110eleq2d 2673 . . 3 (𝑋 ∈ V → (𝐷 ∈ (CMet‘𝑋) ↔ 𝐷 ∈ {𝑑 ∈ (Met‘𝑋) ∣ ∀𝑓 ∈ (CauFil‘𝑑)((MetOpen‘𝑑) fLim 𝑓) ≠ ∅}))
12 fveq2 6103 . . . . 5 (𝑑 = 𝐷 → (CauFil‘𝑑) = (CauFil‘𝐷))
13 fveq2 6103 . . . . . . . 8 (𝑑 = 𝐷 → (MetOpen‘𝑑) = (MetOpen‘𝐷))
14 iscmet.1 . . . . . . . 8 𝐽 = (MetOpen‘𝐷)
1513, 14syl6eqr 2662 . . . . . . 7 (𝑑 = 𝐷 → (MetOpen‘𝑑) = 𝐽)
1615oveq1d 6564 . . . . . 6 (𝑑 = 𝐷 → ((MetOpen‘𝑑) fLim 𝑓) = (𝐽 fLim 𝑓))
1716neeq1d 2841 . . . . 5 (𝑑 = 𝐷 → (((MetOpen‘𝑑) fLim 𝑓) ≠ ∅ ↔ (𝐽 fLim 𝑓) ≠ ∅))
1812, 17raleqbidv 3129 . . . 4 (𝑑 = 𝐷 → (∀𝑓 ∈ (CauFil‘𝑑)((MetOpen‘𝑑) fLim 𝑓) ≠ ∅ ↔ ∀𝑓 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑓) ≠ ∅))
1918elrab 3331 . . 3 (𝐷 ∈ {𝑑 ∈ (Met‘𝑋) ∣ ∀𝑓 ∈ (CauFil‘𝑑)((MetOpen‘𝑑) fLim 𝑓) ≠ ∅} ↔ (𝐷 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑓) ≠ ∅))
2011, 19syl6bb 275 . 2 (𝑋 ∈ V → (𝐷 ∈ (CMet‘𝑋) ↔ (𝐷 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑓) ≠ ∅)))
211, 3, 20pm5.21nii 367 1 (𝐷 ∈ (CMet‘𝑋) ↔ (𝐷 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑓) ≠ ∅))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  {crab 2900  Vcvv 3173  ∅c0 3874  ‘cfv 5804  (class class class)co 6549  Metcme 19553  MetOpencmopn 19557   fLim cflim 21548  CauFilccfil 22858  CMetcms 22860 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-cmet 22863 This theorem is referenced by:  cmetcvg  22891  cmetmet  22892  iscmet3  22899  cmetss  22921  equivcmet  22922  relcmpcmet  22923  cmetcusp1  22957
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