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Theorem cmetmet 23816
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 2818 . . 3 (MetOpen‘𝐷) = (MetOpen‘𝐷)
21iscmet 23814 . 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 2105  wne 3013  wral 3135  c0 4288  cfv 6348  (class class class)co 7145  Metcmet 20459  MetOpencmopn 20463   fLim cflim 22470  CauFilccfil 23782  CMetccmet 23784
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148  df-cmet 23787
This theorem is referenced by:  cmetmeti  23817  cmetcaulem  23818  cmetcau  23819  iscmet2  23824  metsscmetcld  23845  cmetss  23846  bcthlem2  23855  bcthlem3  23856  bcthlem4  23857  bcthlem5  23858  bcth2  23860  bcth3  23861  cmetcusp1  23883  cmetcusp  23884  minveclem3  23959  ubthlem1  28574  ubthlem2  28575  hlmet  28599  fmcncfil  31073  heiborlem3  34972  heiborlem6  34975  heiborlem8  34977  heiborlem9  34978  heiborlem10  34979  heibor  34980  bfplem1  34981  bfplem2  34982  bfp  34983
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