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Theorem isms 24459
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 6906 . . . . 5 (𝑓 = 𝐾 → (dist‘𝑓) = (dist‘𝐾))
2 fveq2 6906 . . . . . . 7 (𝑓 = 𝐾 → (Base‘𝑓) = (Base‘𝐾))
3 isms.x . . . . . . 7 𝑋 = (Base‘𝐾)
42, 3eqtr4di 2795 . . . . . 6 (𝑓 = 𝐾 → (Base‘𝑓) = 𝑋)
54sqxpeqd 5717 . . . . 5 (𝑓 = 𝐾 → ((Base‘𝑓) × (Base‘𝑓)) = (𝑋 × 𝑋))
61, 5reseq12d 5998 . . . 4 (𝑓 = 𝐾 → ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) = ((dist‘𝐾) ↾ (𝑋 × 𝑋)))
7 isms.d . . . 4 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋))
86, 7eqtr4di 2795 . . 3 (𝑓 = 𝐾 → ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) = 𝐷)
94fveq2d 6910 . . 3 (𝑓 = 𝐾 → (Met‘(Base‘𝑓)) = (Met‘𝑋))
108, 9eleq12d 2835 . 2 (𝑓 = 𝐾 → (((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) ∈ (Met‘(Base‘𝑓)) ↔ 𝐷 ∈ (Met‘𝑋)))
11 df-ms 24331 . 2 MetSp = {𝑓 ∈ ∞MetSp ∣ ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) ∈ (Met‘(Base‘𝑓))}
1210, 11elrab2 3695 1 (𝐾 ∈ MetSp ↔ (𝐾 ∈ ∞MetSp ∧ 𝐷 ∈ (Met‘𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  wcel 2108   × cxp 5683  cres 5687  cfv 6561  Basecbs 17247  distcds 17306  TopOpenctopn 17466  Metcmet 21350  ∞MetSpcxms 24327  MetSpcms 24328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-xp 5691  df-res 5697  df-iota 6514  df-fv 6569  df-ms 24331
This theorem is referenced by:  isms2  24460  msxms  24464  mspropd  24484  setsms  24492  tmsms  24500  imasf1oms  24503  ressms  24539  prdsms  24544
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