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Theorem tmsval 22333
Description: For any metric there is an associated metric space. (Contributed by Mario Carneiro, 2-Sep-2015.)
Hypotheses
Ref Expression
tmsval.m 𝑀 = {⟨(Base‘ndx), 𝑋⟩, ⟨(dist‘ndx), 𝐷⟩}
tmsval.k 𝐾 = (toMetSp‘𝐷)
Assertion
Ref Expression
tmsval (𝐷 ∈ (∞Met‘𝑋) → 𝐾 = (𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩))

Proof of Theorem tmsval
Dummy variable 𝑑 is distinct from all other variables.
StepHypRef Expression
1 tmsval.k . 2 𝐾 = (toMetSp‘𝐷)
2 df-tms 22174 . . . 4 toMetSp = (𝑑 ran ∞Met ↦ ({⟨(Base‘ndx), dom dom 𝑑⟩, ⟨(dist‘ndx), 𝑑⟩} sSet ⟨(TopSet‘ndx), (MetOpen‘𝑑)⟩))
32a1i 11 . . 3 (𝐷 ∈ (∞Met‘𝑋) → toMetSp = (𝑑 ran ∞Met ↦ ({⟨(Base‘ndx), dom dom 𝑑⟩, ⟨(dist‘ndx), 𝑑⟩} sSet ⟨(TopSet‘ndx), (MetOpen‘𝑑)⟩)))
4 dmeq 5356 . . . . . . . . 9 (𝑑 = 𝐷 → dom 𝑑 = dom 𝐷)
54dmeqd 5358 . . . . . . . 8 (𝑑 = 𝐷 → dom dom 𝑑 = dom dom 𝐷)
6 xmetf 22181 . . . . . . . . . . 11 (𝐷 ∈ (∞Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
7 fdm 6089 . . . . . . . . . . 11 (𝐷:(𝑋 × 𝑋)⟶ℝ* → dom 𝐷 = (𝑋 × 𝑋))
86, 7syl 17 . . . . . . . . . 10 (𝐷 ∈ (∞Met‘𝑋) → dom 𝐷 = (𝑋 × 𝑋))
98dmeqd 5358 . . . . . . . . 9 (𝐷 ∈ (∞Met‘𝑋) → dom dom 𝐷 = dom (𝑋 × 𝑋))
10 dmxpid 5377 . . . . . . . . 9 dom (𝑋 × 𝑋) = 𝑋
119, 10syl6eq 2701 . . . . . . . 8 (𝐷 ∈ (∞Met‘𝑋) → dom dom 𝐷 = 𝑋)
125, 11sylan9eqr 2707 . . . . . . 7 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → dom dom 𝑑 = 𝑋)
1312opeq2d 4440 . . . . . 6 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → ⟨(Base‘ndx), dom dom 𝑑⟩ = ⟨(Base‘ndx), 𝑋⟩)
14 simpr 476 . . . . . . 7 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → 𝑑 = 𝐷)
1514opeq2d 4440 . . . . . 6 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → ⟨(dist‘ndx), 𝑑⟩ = ⟨(dist‘ndx), 𝐷⟩)
1613, 15preq12d 4308 . . . . 5 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → {⟨(Base‘ndx), dom dom 𝑑⟩, ⟨(dist‘ndx), 𝑑⟩} = {⟨(Base‘ndx), 𝑋⟩, ⟨(dist‘ndx), 𝐷⟩})
17 tmsval.m . . . . 5 𝑀 = {⟨(Base‘ndx), 𝑋⟩, ⟨(dist‘ndx), 𝐷⟩}
1816, 17syl6eqr 2703 . . . 4 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → {⟨(Base‘ndx), dom dom 𝑑⟩, ⟨(dist‘ndx), 𝑑⟩} = 𝑀)
1914fveq2d 6233 . . . . 5 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → (MetOpen‘𝑑) = (MetOpen‘𝐷))
2019opeq2d 4440 . . . 4 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → ⟨(TopSet‘ndx), (MetOpen‘𝑑)⟩ = ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩)
2118, 20oveq12d 6708 . . 3 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → ({⟨(Base‘ndx), dom dom 𝑑⟩, ⟨(dist‘ndx), 𝑑⟩} sSet ⟨(TopSet‘ndx), (MetOpen‘𝑑)⟩) = (𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩))
22 fvssunirn 6255 . . . 4 (∞Met‘𝑋) ⊆ ran ∞Met
2322sseli 3632 . . 3 (𝐷 ∈ (∞Met‘𝑋) → 𝐷 ran ∞Met)
24 ovexd 6720 . . 3 (𝐷 ∈ (∞Met‘𝑋) → (𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩) ∈ V)
253, 21, 23, 24fvmptd 6327 . 2 (𝐷 ∈ (∞Met‘𝑋) → (toMetSp‘𝐷) = (𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩))
261, 25syl5eq 2697 1 (𝐷 ∈ (∞Met‘𝑋) → 𝐾 = (𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  Vcvv 3231  {cpr 4212  cop 4216   cuni 4468  cmpt 4762   × cxp 5141  dom cdm 5143  ran crn 5144  wf 5922  cfv 5926  (class class class)co 6690  *cxr 10111  ndxcnx 15901   sSet csts 15902  Basecbs 15904  TopSetcts 15994  distcds 15997  ∞Metcxmt 19779  MetOpencmopn 19784  toMetSpctmt 22171
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-map 7901  df-xr 10116  df-xmet 19787  df-tms 22174
This theorem is referenced by:  tmslem  22334
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