Theorem totbndmet 38146

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.

Step	Hyp	Ref	Expression
1		istotbnd 38143	. 2 ⊢ (𝑀 ∈ (TotBnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑑 ∈ ℝ+ ∃𝑣 ∈ Fin (∪ 𝑣 = 𝑋 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))))
2	1	simplbi 497	1 ⊢ (𝑀 ∈ (TotBnd‘𝑋) → 𝑀 ∈ (Met‘𝑋))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∀wral 3054  ∃wrex 3064  ∪ cuni 4845  ‘cfv 6492  (class class class)co 7363  Fincfn 8890  ℝ⁺crp 12940  Metcmet 21340  ballcbl 21341  TotBndctotbnd 38140

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 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7366  df-totbnd 38142

This theorem is referenced by:  totbndss  38151  totbndbnd  38163  prdstotbnd  38168