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Theorem isxms 24458
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 6905 . . . 4 (𝑓 = 𝐾 → (TopOpen‘𝑓) = (TopOpen‘𝐾))
2 isms.j . . . 4 𝐽 = (TopOpen‘𝐾)
31, 2eqtr4di 2794 . . 3 (𝑓 = 𝐾 → (TopOpen‘𝑓) = 𝐽)
4 fveq2 6905 . . . . . 6 (𝑓 = 𝐾 → (dist‘𝑓) = (dist‘𝐾))
5 fveq2 6905 . . . . . . . 8 (𝑓 = 𝐾 → (Base‘𝑓) = (Base‘𝐾))
6 isms.x . . . . . . . 8 𝑋 = (Base‘𝐾)
75, 6eqtr4di 2794 . . . . . . 7 (𝑓 = 𝐾 → (Base‘𝑓) = 𝑋)
87sqxpeqd 5716 . . . . . 6 (𝑓 = 𝐾 → ((Base‘𝑓) × (Base‘𝑓)) = (𝑋 × 𝑋))
94, 8reseq12d 5997 . . . . 5 (𝑓 = 𝐾 → ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) = ((dist‘𝐾) ↾ (𝑋 × 𝑋)))
10 isms.d . . . . 5 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋))
119, 10eqtr4di 2794 . . . 4 (𝑓 = 𝐾 → ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) = 𝐷)
1211fveq2d 6909 . . 3 (𝑓 = 𝐾 → (MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓)))) = (MetOpen‘𝐷))
133, 12eqeq12d 2752 . 2 (𝑓 = 𝐾 → ((TopOpen‘𝑓) = (MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓)))) ↔ 𝐽 = (MetOpen‘𝐷)))
14 df-xms 24331 . 2 ∞MetSp = {𝑓 ∈ TopSp ∣ (TopOpen‘𝑓) = (MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))))}
1513, 14elrab2 3694 1 (𝐾 ∈ ∞MetSp ↔ (𝐾 ∈ TopSp ∧ 𝐽 = (MetOpen‘𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1539  wcel 2107   × cxp 5682  cres 5686  cfv 6560  Basecbs 17248  distcds 17307  TopOpenctopn 17467  MetOpencmopn 21355  TopSpctps 22939  ∞MetSpcxms 24328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-xp 5690  df-res 5696  df-iota 6513  df-fv 6568  df-xms 24331
This theorem is referenced by:  isxms2  24459  xmstopn  24462  xmstps  24464  xmspropd  24484
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