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Theorem isms 23045
 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 6651 . . . . 5 (𝑓 = 𝐾 → (dist‘𝑓) = (dist‘𝐾))
2 fveq2 6651 . . . . . . 7 (𝑓 = 𝐾 → (Base‘𝑓) = (Base‘𝐾))
3 isms.x . . . . . . 7 𝑋 = (Base‘𝐾)
42, 3syl6eqr 2877 . . . . . 6 (𝑓 = 𝐾 → (Base‘𝑓) = 𝑋)
54sqxpeqd 5568 . . . . 5 (𝑓 = 𝐾 → ((Base‘𝑓) × (Base‘𝑓)) = (𝑋 × 𝑋))
61, 5reseq12d 5835 . . . 4 (𝑓 = 𝐾 → ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) = ((dist‘𝐾) ↾ (𝑋 × 𝑋)))
7 isms.d . . . 4 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋))
86, 7syl6eqr 2877 . . 3 (𝑓 = 𝐾 → ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) = 𝐷)
94fveq2d 6655 . . 3 (𝑓 = 𝐾 → (Met‘(Base‘𝑓)) = (Met‘𝑋))
108, 9eleq12d 2910 . 2 (𝑓 = 𝐾 → (((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) ∈ (Met‘(Base‘𝑓)) ↔ 𝐷 ∈ (Met‘𝑋)))
11 df-ms 22917 . 2 MetSp = {𝑓 ∈ ∞MetSp ∣ ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) ∈ (Met‘(Base‘𝑓))}
1210, 11elrab2 3668 1 (𝐾 ∈ MetSp ↔ (𝐾 ∈ ∞MetSp ∧ 𝐷 ∈ (Met‘𝑋)))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2115   × cxp 5534   ↾ cres 5538  ‘cfv 6336  Basecbs 16472  distcds 16563  TopOpenctopn 16684  Metcmet 20517  ∞MetSpcxms 22913  MetSpcms 22914 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-rab 3141  df-v 3481  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4820  df-br 5048  df-opab 5110  df-xp 5542  df-res 5548  df-iota 6295  df-fv 6344  df-ms 22917 This theorem is referenced by:  isms2  23046  msxms  23050  mspropd  23070  setsms  23076  tmsms  23083  imasf1oms  23086  ressms  23122  prdsms  23127
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