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Theorem isxms 22984
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 6663 . . . 4 (𝑓 = 𝐾 → (TopOpen‘𝑓) = (TopOpen‘𝐾))
2 isms.j . . . 4 𝐽 = (TopOpen‘𝐾)
31, 2syl6eqr 2871 . . 3 (𝑓 = 𝐾 → (TopOpen‘𝑓) = 𝐽)
4 fveq2 6663 . . . . . 6 (𝑓 = 𝐾 → (dist‘𝑓) = (dist‘𝐾))
5 fveq2 6663 . . . . . . . 8 (𝑓 = 𝐾 → (Base‘𝑓) = (Base‘𝐾))
6 isms.x . . . . . . . 8 𝑋 = (Base‘𝐾)
75, 6syl6eqr 2871 . . . . . . 7 (𝑓 = 𝐾 → (Base‘𝑓) = 𝑋)
87sqxpeqd 5580 . . . . . 6 (𝑓 = 𝐾 → ((Base‘𝑓) × (Base‘𝑓)) = (𝑋 × 𝑋))
94, 8reseq12d 5847 . . . . 5 (𝑓 = 𝐾 → ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) = ((dist‘𝐾) ↾ (𝑋 × 𝑋)))
10 isms.d . . . . 5 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋))
119, 10syl6eqr 2871 . . . 4 (𝑓 = 𝐾 → ((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))) = 𝐷)
1211fveq2d 6667 . . 3 (𝑓 = 𝐾 → (MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓)))) = (MetOpen‘𝐷))
133, 12eqeq12d 2834 . 2 (𝑓 = 𝐾 → ((TopOpen‘𝑓) = (MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓)))) ↔ 𝐽 = (MetOpen‘𝐷)))
14 df-xms 22857 . 2 ∞MetSp = {𝑓 ∈ TopSp ∣ (TopOpen‘𝑓) = (MetOpen‘((dist‘𝑓) ↾ ((Base‘𝑓) × (Base‘𝑓))))}
1513, 14elrab2 3680 1 (𝐾 ∈ ∞MetSp ↔ (𝐾 ∈ TopSp ∧ 𝐽 = (MetOpen‘𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396   = wceq 1528  wcel 2105   × cxp 5546  cres 5550  cfv 6348  Basecbs 16471  distcds 16562  TopOpenctopn 16683  MetOpencmopn 20463  TopSpctps 21468  ∞MetSpcxms 22854
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-xp 5554  df-res 5560  df-iota 6307  df-fv 6356  df-xms 22857
This theorem is referenced by:  isxms2  22985  xmstopn  22988  xmstps  22990  xmspropd  23010
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