Theorem bndmet 35967

Description: A bounded metric space is a metric space. (Contributed by Mario Carneiro, 16-Sep-2015.)

Assertion

Ref	Expression
bndmet	⊢ (𝑀 ∈ (Bnd‘𝑋) → 𝑀 ∈ (Met‘𝑋))

Proof of Theorem bndmet

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isbnd 35966	. 2 ⊢ (𝑀 ∈ (Bnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑥 ∈ 𝑋 ∃𝑦 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑦)))
2	1	simplbi 497	1 ⊢ (𝑀 ∈ (Bnd‘𝑋) → 𝑀 ∈ (Met‘𝑋))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2101  ∀wral 3059  ∃wrex 3068  ‘cfv 6447  (class class class)co 7295  ℝ⁺crp 12758  Metcmet 20611  ballcbl 20612  Bndcbnd 35953

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2103  ax-9 2111  ax-10 2132  ax-11 2149  ax-12 2166  ax-ext 2704  ax-sep 5226  ax-nul 5233  ax-pr 5355

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2063  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2884  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3224  df-v 3436  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4260  df-if 4463  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4842  df-br 5078  df-opab 5140  df-mpt 5161  df-id 5491  df-xp 5597  df-rel 5598  df-cnv 5599  df-co 5600  df-dm 5601  df-iota 6399  df-fun 6449  df-fv 6455  df-ov 7298  df-bnd 35965

This theorem is referenced by:  isbnd3  35970  equivbnd  35976  bnd2lem  35977  equivbnd2  35978  prdsbnd  35979  prdsbnd2  35981