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Theorem isms 22194
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 6158 . . . . 5 (𝑓 = 𝐾 → (dist‘𝑓) = (dist‘𝐾))
2 fveq2 6158 . . . . . . 7 (𝑓 = 𝐾 → (Base‘𝑓) = (Base‘𝐾))
3 isms.x . . . . . . 7 𝑋 = (Base‘𝐾)
42, 3syl6eqr 2673 . . . . . 6 (𝑓 = 𝐾 → (Base‘𝑓) = 𝑋)
54sqxpeqd 5111 . . . . 5 (𝑓 = 𝐾 → ((Base‘𝑓) × (Base‘𝑓)) = (𝑋 × 𝑋))
61, 5reseq12d 5367 . . . 4 (𝑓 = 𝐾 → ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) = ((dist‘𝐾) ↾ (𝑋 × 𝑋)))
7 isms.d . . . 4 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋))
86, 7syl6eqr 2673 . . 3 (𝑓 = 𝐾 → ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) = 𝐷)
94fveq2d 6162 . . 3 (𝑓 = 𝐾 → (Met‘(Base‘𝑓)) = (Met‘𝑋))
108, 9eleq12d 2692 . 2 (𝑓 = 𝐾 → (((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) ∈ (Met‘(Base‘𝑓)) ↔ 𝐷 ∈ (Met‘𝑋)))
11 df-ms 22066 . 2 MetSp = {𝑓 ∈ ∞MetSp ∣ ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) ∈ (Met‘(Base‘𝑓))}
1210, 11elrab2 3353 1 (𝐾 ∈ MetSp ↔ (𝐾 ∈ ∞MetSp ∧ 𝐷 ∈ (Met‘𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1480  wcel 1987   × cxp 5082  cres 5086  cfv 5857  Basecbs 15800  distcds 15890  TopOpenctopn 16022  Metcme 19672  ∞MetSpcxme 22062  MetSpcmt 22063
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-ms 22066
This theorem is referenced by:  isms2  22195  msxms  22199  mspropd  22219  setsms  22225  tmsms  22232  imasf1oms  22235  ressms  22271  prdsms  22276
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