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Theorem cmetmet 24673
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 2733 . . 3 (MetOpenβ€˜π·) = (MetOpenβ€˜π·)
21iscmet 24671 . 2 (𝐷 ∈ (CMetβ€˜π‘‹) ↔ (𝐷 ∈ (Metβ€˜π‘‹) ∧ βˆ€π‘“ ∈ (CauFilβ€˜π·)((MetOpenβ€˜π·) fLim 𝑓) β‰  βˆ…))
32simplbi 499 1 (𝐷 ∈ (CMetβ€˜π‘‹) β†’ 𝐷 ∈ (Metβ€˜π‘‹))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∈ wcel 2107   β‰  wne 2940  βˆ€wral 3061  βˆ…c0 4286  β€˜cfv 6500  (class class class)co 7361  Metcmet 20805  MetOpencmopn 20809   fLim cflim 23308  CauFilccfil 24639  CMetccmet 24641
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-iota 6452  df-fun 6502  df-fv 6508  df-ov 7364  df-cmet 24644
This theorem is referenced by:  cmetmeti  24674  cmetcaulem  24675  cmetcau  24676  iscmet2  24681  metsscmetcld  24702  cmetss  24703  bcthlem2  24712  bcthlem3  24713  bcthlem4  24714  bcthlem5  24715  bcth2  24717  bcth3  24718  cmetcusp1  24740  cmetcusp  24741  minveclem3  24816  ubthlem1  29861  ubthlem2  29862  hlmet  29886  fmcncfil  32576  heiborlem3  36322  heiborlem6  36325  heiborlem8  36327  heiborlem9  36328  heiborlem10  36329  heibor  36330  bfplem1  36331  bfplem2  36332  bfp  36333
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