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Theorem cmetmet 24450
Description: A complete metric space is a metric space. (Contributed by NM, 18-Dec-2006.) (Revised by Mario Carneiro, 29-Jan-2014.)
Assertion
Ref Expression
cmetmet (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))

Proof of Theorem cmetmet
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 eqid 2738 . . 3 (MetOpen‘𝐷) = (MetOpen‘𝐷)
21iscmet 24448 . 2 (𝐷 ∈ (CMet‘𝑋) ↔ (𝐷 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐷)((MetOpen‘𝐷) fLim 𝑓) ≠ ∅))
32simplbi 498 1 (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  wne 2943  wral 3064  c0 4256  cfv 6433  (class class class)co 7275  Metcmet 20583  MetOpencmopn 20587   fLim cflim 23085  CauFilccfil 24416  CMetccmet 24418
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441  df-ov 7278  df-cmet 24421
This theorem is referenced by:  cmetmeti  24451  cmetcaulem  24452  cmetcau  24453  iscmet2  24458  metsscmetcld  24479  cmetss  24480  bcthlem2  24489  bcthlem3  24490  bcthlem4  24491  bcthlem5  24492  bcth2  24494  bcth3  24495  cmetcusp1  24517  cmetcusp  24518  minveclem3  24593  ubthlem1  29232  ubthlem2  29233  hlmet  29257  fmcncfil  31881  heiborlem3  35971  heiborlem6  35974  heiborlem8  35976  heiborlem9  35977  heiborlem10  35978  heibor  35979  bfplem1  35980  bfplem2  35981  bfp  35982
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