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Theorem bndmet 26481
 Description: A bounded metric space is a metric space. (Contributed by Mario Carneiro, 16-Sep-2015.)
Assertion
Ref Expression
bndmet

Proof of Theorem bndmet
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isbnd 26480 . 2
21simplbi 447 1
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1652   wcel 1725  wral 2697  wrex 2698  cfv 5446  (class class class)co 6073  crp 10604  cme 16679  cbl 16680  cbnd 26467 This theorem is referenced by:  isbnd3  26484  equivbnd  26490  bnd2lem  26491  equivbnd2  26492  prdsbnd  26493  prdsbnd2  26495 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-bnd 26479
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