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Theorem tmsval 23989
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 23828 . . 3 toMetSp = (𝑑 ran ∞Met ↦ ({⟨(Base‘ndx), dom dom 𝑑⟩, ⟨(dist‘ndx), 𝑑⟩} sSet ⟨(TopSet‘ndx), (MetOpen‘𝑑)⟩))
3 dmeq 5904 . . . . . . . . 9 (𝑑 = 𝐷 → dom 𝑑 = dom 𝐷)
43dmeqd 5906 . . . . . . . 8 (𝑑 = 𝐷 → dom dom 𝑑 = dom dom 𝐷)
5 xmetf 23835 . . . . . . . . . . 11 (𝐷 ∈ (∞Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
65fdmd 6729 . . . . . . . . . 10 (𝐷 ∈ (∞Met‘𝑋) → dom 𝐷 = (𝑋 × 𝑋))
76dmeqd 5906 . . . . . . . . 9 (𝐷 ∈ (∞Met‘𝑋) → dom dom 𝐷 = dom (𝑋 × 𝑋))
8 dmxpid 5930 . . . . . . . . 9 dom (𝑋 × 𝑋) = 𝑋
97, 8eqtrdi 2789 . . . . . . . 8 (𝐷 ∈ (∞Met‘𝑋) → dom dom 𝐷 = 𝑋)
104, 9sylan9eqr 2795 . . . . . . 7 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → dom dom 𝑑 = 𝑋)
1110opeq2d 4881 . . . . . 6 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → ⟨(Base‘ndx), dom dom 𝑑⟩ = ⟨(Base‘ndx), 𝑋⟩)
12 simpr 486 . . . . . . 7 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → 𝑑 = 𝐷)
1312opeq2d 4881 . . . . . 6 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → ⟨(dist‘ndx), 𝑑⟩ = ⟨(dist‘ndx), 𝐷⟩)
1411, 13preq12d 4746 . . . . 5 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → {⟨(Base‘ndx), dom dom 𝑑⟩, ⟨(dist‘ndx), 𝑑⟩} = {⟨(Base‘ndx), 𝑋⟩, ⟨(dist‘ndx), 𝐷⟩})
15 tmsval.m . . . . 5 𝑀 = {⟨(Base‘ndx), 𝑋⟩, ⟨(dist‘ndx), 𝐷⟩}
1614, 15eqtr4di 2791 . . . 4 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → {⟨(Base‘ndx), dom dom 𝑑⟩, ⟨(dist‘ndx), 𝑑⟩} = 𝑀)
1712fveq2d 6896 . . . . 5 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → (MetOpen‘𝑑) = (MetOpen‘𝐷))
1817opeq2d 4881 . . . 4 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → ⟨(TopSet‘ndx), (MetOpen‘𝑑)⟩ = ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩)
1916, 18oveq12d 7427 . . 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)⟩))
20 fvssunirn 6925 . . . 4 (∞Met‘𝑋) ⊆ ran ∞Met
2120sseli 3979 . . 3 (𝐷 ∈ (∞Met‘𝑋) → 𝐷 ran ∞Met)
22 ovexd 7444 . . 3 (𝐷 ∈ (∞Met‘𝑋) → (𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩) ∈ V)
232, 19, 21, 22fvmptd2 7007 . 2 (𝐷 ∈ (∞Met‘𝑋) → (toMetSp‘𝐷) = (𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩))
241, 23eqtrid 2785 1 (𝐷 ∈ (∞Met‘𝑋) → 𝐾 = (𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  Vcvv 3475  {cpr 4631  cop 4635   cuni 4909   × cxp 5675  dom cdm 5677  ran crn 5678  cfv 6544  (class class class)co 7409  *cxr 11247   sSet csts 17096  ndxcnx 17126  Basecbs 17144  TopSetcts 17203  distcds 17206  ∞Metcxmet 20929  MetOpencmopn 20934  toMetSpctms 23825
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-map 8822  df-xr 11252  df-xmet 20937  df-tms 23828
This theorem is referenced by:  tmslem  23990  tmslemOLD  23991
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