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Theorem totbndmet 33551
Description: The predicate "totally bounded" implies 𝑀 is a metric space. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
totbndmet (𝑀 ∈ (TotBnd‘𝑋) → 𝑀 ∈ (Met‘𝑋))

Proof of Theorem totbndmet
Dummy variables 𝑏 𝑑 𝑣 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 istotbnd 33548 . 2 (𝑀 ∈ (TotBnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑑 ∈ ℝ+𝑣 ∈ Fin ( 𝑣 = 𝑋 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))))
21simplbi 476 1 (𝑀 ∈ (TotBnd‘𝑋) → 𝑀 ∈ (Met‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1482  wcel 1989  wral 2911  wrex 2912   cuni 4434  cfv 5886  (class class class)co 6647  Fincfn 7952  +crp 11829  Metcme 19726  ballcbl 19727  TotBndctotbnd 33545
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-iota 5849  df-fun 5888  df-fv 5894  df-ov 6650  df-totbnd 33547
This theorem is referenced by:  totbndss  33556  totbndbnd  33568  prdstotbnd  33573
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