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Theorem cmetmeti 24647
Description: A complete metric space is a metric space. (Contributed by NM, 26-Oct-2007.)
Hypothesis
Ref Expression
cmetmeti.1 𝐷 ∈ (CMet‘𝑋)
Assertion
Ref Expression
cmetmeti 𝐷 ∈ (Met‘𝑋)

Proof of Theorem cmetmeti
StepHypRef Expression
1 cmetmeti.1 . 2 𝐷 ∈ (CMet‘𝑋)
2 cmetmet 24646 . 2 (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
31, 2ax-mp 5 1 𝐷 ∈ (Met‘𝑋)
Colors of variables: wff setvar class
Syntax hints:  wcel 2106  cfv 6494  Metcmet 20778  CMetccmet 24614
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5255  ax-nul 5262  ax-pr 5383
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-br 5105  df-opab 5167  df-mpt 5188  df-id 5530  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-iota 6446  df-fun 6496  df-fv 6502  df-ov 7357  df-cmet 24617
This theorem is referenced by: (None)
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