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Theorem isms 22474
Description: Express the predicate "𝑋, 𝐷 is a metric space" with underlying set 𝑋 and distance function 𝐷. (Contributed by NM, 27-Aug-2006.) (Revised by Mario Carneiro, 24-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
isms (𝐾 ∈ MetSp ↔ (𝐾 ∈ ∞MetSp ∧ 𝐷 ∈ (Met‘𝑋)))

Proof of Theorem isms
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6332 . . . . 5 (𝑓 = 𝐾 → (dist‘𝑓) = (dist‘𝐾))
2 fveq2 6332 . . . . . . 7 (𝑓 = 𝐾 → (Base‘𝑓) = (Base‘𝐾))
3 isms.x . . . . . . 7 𝑋 = (Base‘𝐾)
42, 3syl6eqr 2823 . . . . . 6 (𝑓 = 𝐾 → (Base‘𝑓) = 𝑋)
54sqxpeqd 5281 . . . . 5 (𝑓 = 𝐾 → ((Base‘𝑓) × (Base‘𝑓)) = (𝑋 × 𝑋))
61, 5reseq12d 5535 . . . 4 (𝑓 = 𝐾 → ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) = ((dist‘𝐾) ↾ (𝑋 × 𝑋)))
7 isms.d . . . 4 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋))
86, 7syl6eqr 2823 . . 3 (𝑓 = 𝐾 → ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) = 𝐷)
94fveq2d 6336 . . 3 (𝑓 = 𝐾 → (Met‘(Base‘𝑓)) = (Met‘𝑋))
108, 9eleq12d 2844 . 2 (𝑓 = 𝐾 → (((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) ∈ (Met‘(Base‘𝑓)) ↔ 𝐷 ∈ (Met‘𝑋)))
11 df-ms 22346 . 2 MetSp = {𝑓 ∈ ∞MetSp ∣ ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) ∈ (Met‘(Base‘𝑓))}
1210, 11elrab2 3518 1 (𝐾 ∈ MetSp ↔ (𝐾 ∈ ∞MetSp ∧ 𝐷 ∈ (Met‘𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382   = wceq 1631  wcel 2145   × cxp 5247  cres 5251  cfv 6031  Basecbs 16064  distcds 16158  TopOpenctopn 16290  Metcme 19947  ∞MetSpcxme 22342  MetSpcmt 22343
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-xp 5255  df-res 5261  df-iota 5994  df-fv 6039  df-ms 22346
This theorem is referenced by:  isms2  22475  msxms  22479  mspropd  22499  setsms  22505  tmsms  22512  imasf1oms  22515  ressms  22551  prdsms  22556
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