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Theorem iscmet 25331
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 6944 . 2 (𝐷 ∈ (CMet‘𝑋) → 𝑋 ∈ V)
2 elfvex 6944 . . 3 (𝐷 ∈ (Met‘𝑋) → 𝑋 ∈ V)
32adantr 480 . 2 ((𝐷 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑓) ≠ ∅) → 𝑋 ∈ V)
4 fveq2 6906 . . . . . 6 (𝑥 = 𝑋 → (Met‘𝑥) = (Met‘𝑋))
54rabeqdv 3448 . . . . 5 (𝑥 = 𝑋 → {𝑑 ∈ (Met‘𝑥) ∣ ∀𝑓 ∈ (CauFil‘𝑑)((MetOpen‘𝑑) fLim 𝑓) ≠ ∅} = {𝑑 ∈ (Met‘𝑋) ∣ ∀𝑓 ∈ (CauFil‘𝑑)((MetOpen‘𝑑) fLim 𝑓) ≠ ∅})
6 df-cmet 25304 . . . . 5 CMet = (𝑥 ∈ V ↦ {𝑑 ∈ (Met‘𝑥) ∣ ∀𝑓 ∈ (CauFil‘𝑑)((MetOpen‘𝑑) fLim 𝑓) ≠ ∅})
7 fvex 6919 . . . . . 6 (Met‘𝑋) ∈ V
87rabex 5344 . . . . 5 {𝑑 ∈ (Met‘𝑋) ∣ ∀𝑓 ∈ (CauFil‘𝑑)((MetOpen‘𝑑) fLim 𝑓) ≠ ∅} ∈ V
95, 6, 8fvmpt 7015 . . . 4 (𝑋 ∈ V → (CMet‘𝑋) = {𝑑 ∈ (Met‘𝑋) ∣ ∀𝑓 ∈ (CauFil‘𝑑)((MetOpen‘𝑑) fLim 𝑓) ≠ ∅})
109eleq2d 2824 . . 3 (𝑋 ∈ V → (𝐷 ∈ (CMet‘𝑋) ↔ 𝐷 ∈ {𝑑 ∈ (Met‘𝑋) ∣ ∀𝑓 ∈ (CauFil‘𝑑)((MetOpen‘𝑑) fLim 𝑓) ≠ ∅}))
11 fveq2 6906 . . . . 5 (𝑑 = 𝐷 → (CauFil‘𝑑) = (CauFil‘𝐷))
12 fveq2 6906 . . . . . . . 8 (𝑑 = 𝐷 → (MetOpen‘𝑑) = (MetOpen‘𝐷))
13 iscmet.1 . . . . . . . 8 𝐽 = (MetOpen‘𝐷)
1412, 13eqtr4di 2792 . . . . . . 7 (𝑑 = 𝐷 → (MetOpen‘𝑑) = 𝐽)
1514oveq1d 7445 . . . . . 6 (𝑑 = 𝐷 → ((MetOpen‘𝑑) fLim 𝑓) = (𝐽 fLim 𝑓))
1615neeq1d 2997 . . . . 5 (𝑑 = 𝐷 → (((MetOpen‘𝑑) fLim 𝑓) ≠ ∅ ↔ (𝐽 fLim 𝑓) ≠ ∅))
1711, 16raleqbidv 3343 . . . 4 (𝑑 = 𝐷 → (∀𝑓 ∈ (CauFil‘𝑑)((MetOpen‘𝑑) fLim 𝑓) ≠ ∅ ↔ ∀𝑓 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑓) ≠ ∅))
1817elrab 3694 . . 3 (𝐷 ∈ {𝑑 ∈ (Met‘𝑋) ∣ ∀𝑓 ∈ (CauFil‘𝑑)((MetOpen‘𝑑) fLim 𝑓) ≠ ∅} ↔ (𝐷 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑓) ≠ ∅))
1910, 18bitrdi 287 . 2 (𝑋 ∈ V → (𝐷 ∈ (CMet‘𝑋) ↔ (𝐷 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑓) ≠ ∅)))
201, 3, 19pm5.21nii 378 1 (𝐷 ∈ (CMet‘𝑋) ↔ (𝐷 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑓) ≠ ∅))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1536  wcel 2105  wne 2937  wral 3058  {crab 3432  Vcvv 3477  c0 4338  cfv 6562  (class class class)co 7430  Metcmet 21367  MetOpencmopn 21371   fLim cflim 23957  CauFilccfil 25299  CMetccmet 25301
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-iota 6515  df-fun 6564  df-fv 6570  df-ov 7433  df-cmet 25304
This theorem is referenced by:  cmetcvg  25332  cmetmet  25333  iscmet3  25340  cmetss  25363  equivcmet  25364  relcmpcmet  25365  cmetcusp1  25400
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