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Theorem xmstopn 22196
Description: The topology component of a metric space coincides with the topology generated by the metric component. (Contributed by Mario Carneiro, 26-Aug-2015.)
Hypotheses
Ref Expression
isms.j 𝐽 = (TopOpen‘𝐾)
isms.x 𝑋 = (Base‘𝐾)
isms.d 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋))
Assertion
Ref Expression
xmstopn (𝐾 ∈ ∞MetSp → 𝐽 = (MetOpen‘𝐷))

Proof of Theorem xmstopn
StepHypRef Expression
1 isms.j . . 3 𝐽 = (TopOpen‘𝐾)
2 isms.x . . 3 𝑋 = (Base‘𝐾)
3 isms.d . . 3 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋))
41, 2, 3isxms 22192 . 2 (𝐾 ∈ ∞MetSp ↔ (𝐾 ∈ TopSp ∧ 𝐽 = (MetOpen‘𝐷)))
54simprbi 480 1 (𝐾 ∈ ∞MetSp → 𝐽 = (MetOpen‘𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987   × cxp 5082  cres 5086  cfv 5857  Basecbs 15800  distcds 15890  TopOpenctopn 16022  MetOpencmopn 19676  TopSpctps 20676  ∞MetSpcxme 22062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-xp 5090  df-res 5096  df-iota 5820  df-fv 5865  df-xms 22065
This theorem is referenced by:  imasf1oxms  22234  ressxms  22270  prdsxmslem2  22274  tmsxpsmopn  22282  xmsusp  22314  cmetcusp1  23089  minveclem4a  23141  minveclem4  23143  qqhcn  29859  rrhcn  29865  rrexthaus  29875  dya2icoseg2  30163  sitmcl  30236
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