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Theorem isxms 24473
Description: Express the predicate "𝑋, 𝐷 is an extended metric space" with underlying set 𝑋 and distance function 𝐷. (Contributed by Mario Carneiro, 2-Sep-2015.)
Hypotheses
Ref Expression
isms.j 𝐽 = (TopOpen‘𝐾)
isms.x 𝑋 = (Base‘𝐾)
isms.d 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋))
Assertion
Ref Expression
isxms (𝐾 ∈ ∞MetSp ↔ (𝐾 ∈ TopSp ∧ 𝐽 = (MetOpen‘𝐷)))

Proof of Theorem isxms
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6907 . . . 4 (𝑓 = 𝐾 → (TopOpen‘𝑓) = (TopOpen‘𝐾))
2 isms.j . . . 4 𝐽 = (TopOpen‘𝐾)
31, 2eqtr4di 2793 . . 3 (𝑓 = 𝐾 → (TopOpen‘𝑓) = 𝐽)
4 fveq2 6907 . . . . . 6 (𝑓 = 𝐾 → (dist‘𝑓) = (dist‘𝐾))
5 fveq2 6907 . . . . . . . 8 (𝑓 = 𝐾 → (Base‘𝑓) = (Base‘𝐾))
6 isms.x . . . . . . . 8 𝑋 = (Base‘𝐾)
75, 6eqtr4di 2793 . . . . . . 7 (𝑓 = 𝐾 → (Base‘𝑓) = 𝑋)
87sqxpeqd 5721 . . . . . 6 (𝑓 = 𝐾 → ((Base‘𝑓) × (Base‘𝑓)) = (𝑋 × 𝑋))
94, 8reseq12d 6001 . . . . 5 (𝑓 = 𝐾 → ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) = ((dist‘𝐾) ↾ (𝑋 × 𝑋)))
10 isms.d . . . . 5 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋))
119, 10eqtr4di 2793 . . . 4 (𝑓 = 𝐾 → ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) = 𝐷)
1211fveq2d 6911 . . 3 (𝑓 = 𝐾 → (MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓)))) = (MetOpen‘𝐷))
133, 12eqeq12d 2751 . 2 (𝑓 = 𝐾 → ((TopOpen‘𝑓) = (MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓)))) ↔ 𝐽 = (MetOpen‘𝐷)))
14 df-xms 24346 . 2 ∞MetSp = {𝑓 ∈ TopSp ∣ (TopOpen‘𝑓) = (MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))))}
1513, 14elrab2 3698 1 (𝐾 ∈ ∞MetSp ↔ (𝐾 ∈ TopSp ∧ 𝐽 = (MetOpen‘𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1537  wcel 2106   × cxp 5687  cres 5691  cfv 6563  Basecbs 17245  distcds 17307  TopOpenctopn 17468  MetOpencmopn 21372  TopSpctps 22954  ∞MetSpcxms 24343
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-res 5701  df-iota 6516  df-fv 6571  df-xms 24346
This theorem is referenced by:  isxms2  24474  xmstopn  24477  xmstps  24479  xmspropd  24499
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