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Theorem cmetmet 23302
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 2770 . . 3 (MetOpen‘𝐷) = (MetOpen‘𝐷)
21iscmet 23300 . 2 (𝐷 ∈ (CMet‘𝑋) ↔ (𝐷 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐷)((MetOpen‘𝐷) fLim 𝑓) ≠ ∅))
32simplbi 479 1 (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2144  wne 2942  wral 3060  c0 4061  cfv 6031  (class class class)co 6792  Metcme 19946  MetOpencmopn 19950   fLim cflim 21957  CauFilccfil 23268  CMetcms 23270
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6795  df-cmet 23273
This theorem is referenced by:  cmetmeti  23303  cmetcaulem  23304  cmetcau  23305  iscmet2  23310  cmetss  23331  bcthlem2  23340  bcthlem3  23341  bcthlem4  23342  bcthlem5  23343  bcth2  23345  bcth3  23346  cmetcusp1  23367  cmetcusp  23368  minveclem3  23418  ubthlem1  28060  ubthlem2  28061  hlmet  28085  fmcncfil  30311  heiborlem3  33937  heiborlem6  33940  heiborlem8  33942  heiborlem9  33943  heiborlem10  33944  heibor  33945  bfplem1  33946  bfplem2  33947  bfp  33948
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