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Theorem cmetmet 22806
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 2605 . . 3 (MetOpen‘𝐷) = (MetOpen‘𝐷)
21iscmet 22804 . 2 (𝐷 ∈ (CMet‘𝑋) ↔ (𝐷 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐷)((MetOpen‘𝐷) fLim 𝑓) ≠ ∅))
32simplbi 474 1 (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1975  wne 2775  wral 2891  c0 3869  cfv 5786  (class class class)co 6523  Metcme 19495  MetOpencmopn 19499   fLim cflim 21486  CauFilccfil 22772  CMetcms 22774
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-rab 2900  df-v 3170  df-sbc 3398  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-br 4574  df-opab 4634  df-mpt 4635  df-id 4939  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-iota 5750  df-fun 5788  df-fv 5794  df-ov 6526  df-cmet 22777
This theorem is referenced by:  cmetmeti  22807  cmetcaulem  22808  cmetcau  22809  iscmet2  22814  cmetss  22834  bcthlem2  22843  bcthlem3  22844  bcthlem4  22845  bcthlem5  22846  bcth2  22848  bcth3  22849  cmetcusp1  22870  cmetcusp  22871  minveclem3  22921  ubthlem1  26912  ubthlem2  26913  hlmet  26937  fmcncfil  29107  heiborlem3  32581  heiborlem6  32584  heiborlem8  32586  heiborlem9  32587  heiborlem10  32588  heibor  32589  bfplem1  32590  bfplem2  32591  bfp  32592
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