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Theorem cmetmet 23881
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 2819 . . 3 (MetOpen‘𝐷) = (MetOpen‘𝐷)
21iscmet 23879 . 2 (𝐷 ∈ (CMet‘𝑋) ↔ (𝐷 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐷)((MetOpen‘𝐷) fLim 𝑓) ≠ ∅))
32simplbi 500 1 (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  wne 3014  wral 3136  c0 4289  cfv 6348  (class class class)co 7148  Metcmet 20523  MetOpencmopn 20527   fLim cflim 22534  CauFilccfil 23847  CMetccmet 23849
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  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 7151  df-cmet 23852
This theorem is referenced by:  cmetmeti  23882  cmetcaulem  23883  cmetcau  23884  iscmet2  23889  metsscmetcld  23910  cmetss  23911  bcthlem2  23920  bcthlem3  23921  bcthlem4  23922  bcthlem5  23923  bcth2  23925  bcth3  23926  cmetcusp1  23948  cmetcusp  23949  minveclem3  24024  ubthlem1  28639  ubthlem2  28640  hlmet  28664  fmcncfil  31167  heiborlem3  35083  heiborlem6  35086  heiborlem8  35088  heiborlem9  35089  heiborlem10  35090  heibor  35091  bfplem1  35092  bfplem2  35093  bfp  35094
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