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Theorem cmetmet 24794
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 2732 . . 3 (MetOpen‘𝐷) = (MetOpen‘𝐷)
21iscmet 24792 . 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 2940  wral 3061  c0 4321  cfv 6540  (class class class)co 7405  Metcmet 20922  MetOpencmopn 20926   fLim cflim 23429  CauFilccfil 24760  CMetccmet 24762
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6492  df-fun 6542  df-fv 6548  df-ov 7408  df-cmet 24765
This theorem is referenced by:  cmetmeti  24795  cmetcaulem  24796  cmetcau  24797  iscmet2  24802  metsscmetcld  24823  cmetss  24824  bcthlem2  24833  bcthlem3  24834  bcthlem4  24835  bcthlem5  24836  bcth2  24838  bcth3  24839  cmetcusp1  24861  cmetcusp  24862  minveclem3  24937  ubthlem1  30110  ubthlem2  30111  hlmet  30135  fmcncfil  32899  heiborlem3  36669  heiborlem6  36672  heiborlem8  36674  heiborlem9  36675  heiborlem10  36676  heibor  36677  bfplem1  36678  bfplem2  36679  bfp  36680
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