Theorem totbndmet 34603

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 34600	. 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 498	1 ⊢ (𝑀 ∈ (TotBnd‘𝑋) → 𝑀 ∈ (Met‘𝑋))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1525   ∈ wcel 2083  ∀wral 3107  ∃wrex 3108  ∪ cuni 4751  ‘cfv 6232  (class class class)co 7023  Fincfn 8364  ℝ⁺crp 12243  Metcmet 20217  ballcbl 20218  TotBndctotbnd 34597

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-sbc 3712  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-br 4969  df-opab 5031  df-mpt 5048  df-id 5355  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-iota 6196  df-fun 6234  df-fv 6240  df-ov 7026  df-totbnd 34599

This theorem is referenced by:  totbndss  34608  totbndbnd  34620  prdstotbnd  34625