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Theorem cmetmet 24038
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 2738 . . 3 (MetOpen‘𝐷) = (MetOpen‘𝐷)
21iscmet 24036 . 2 (𝐷 ∈ (CMet‘𝑋) ↔ (𝐷 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐷)((MetOpen‘𝐷) fLim 𝑓) ≠ ∅))
32simplbi 501 1 (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2114  wne 2934  wral 3053  c0 4211  cfv 6339  (class class class)co 7170  Metcmet 20203  MetOpencmopn 20207   fLim cflim 22685  CauFilccfil 24004  CMetccmet 24006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3681  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-iota 6297  df-fun 6341  df-fv 6347  df-ov 7173  df-cmet 24009
This theorem is referenced by:  cmetmeti  24039  cmetcaulem  24040  cmetcau  24041  iscmet2  24046  metsscmetcld  24067  cmetss  24068  bcthlem2  24077  bcthlem3  24078  bcthlem4  24079  bcthlem5  24080  bcth2  24082  bcth3  24083  cmetcusp1  24105  cmetcusp  24106  minveclem3  24181  ubthlem1  28805  ubthlem2  28806  hlmet  28830  fmcncfil  31453  heiborlem3  35594  heiborlem6  35597  heiborlem8  35599  heiborlem9  35600  heiborlem10  35601  heibor  35602  bfplem1  35603  bfplem2  35604  bfp  35605
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