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Theorem cmetmet 25027
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 ∈ (CMetβ€˜π‘‹) β†’ 𝐷 ∈ (Metβ€˜π‘‹))

Proof of Theorem cmetmet
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 eqid 2732 . . 3 (MetOpenβ€˜π·) = (MetOpenβ€˜π·)
21iscmet 25025 . 2 (𝐷 ∈ (CMetβ€˜π‘‹) ↔ (𝐷 ∈ (Metβ€˜π‘‹) ∧ βˆ€π‘“ ∈ (CauFilβ€˜π·)((MetOpenβ€˜π·) fLim 𝑓) β‰  βˆ…))
32simplbi 498 1 (𝐷 ∈ (CMetβ€˜π‘‹) β†’ 𝐷 ∈ (Metβ€˜π‘‹))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∈ wcel 2106   β‰  wne 2940  βˆ€wral 3061  βˆ…c0 4322  β€˜cfv 6543  (class class class)co 7411  Metcmet 21130  MetOpencmopn 21134   fLim cflim 23658  CauFilccfil 24993  CMetccmet 24995
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 5299  ax-nul 5306  ax-pr 5427
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 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7414  df-cmet 24998
This theorem is referenced by:  cmetmeti  25028  cmetcaulem  25029  cmetcau  25030  iscmet2  25035  metsscmetcld  25056  cmetss  25057  bcthlem2  25066  bcthlem3  25067  bcthlem4  25068  bcthlem5  25069  bcth2  25071  bcth3  25072  cmetcusp1  25094  cmetcusp  25095  minveclem3  25170  ubthlem1  30378  ubthlem2  30379  hlmet  30403  fmcncfil  33197  heiborlem3  36984  heiborlem6  36987  heiborlem8  36989  heiborlem9  36990  heiborlem10  36991  heibor  36992  bfplem1  36993  bfplem2  36994  bfp  36995
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