Theorem tmsval 24649

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.

Step	Hyp	Ref	Expression
1		tmsval.k	. 2 ⊢ 𝐾 = (toMetSp‘𝐷)
2		df-tms 24490	. . 3 ⊢ toMetSp = (𝑑 ∈ ∪ ran ∞Met ↦ ({⟨(Base‘ndx), dom dom 𝑑⟩, ⟨(dist‘ndx), 𝑑⟩} sSet ⟨(TopSet‘ndx), (MetOpen‘𝑑)⟩))
3		dmeq 5892	. . . . . . . . 9 ⊢ (𝑑 = 𝐷 → dom 𝑑 = dom 𝐷)
4	3	dmeqd 5894	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → dom dom 𝑑 = dom dom 𝐷)
5		xmetf 24497	. . . . . . . . . . 11 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
6	5	fdmd 6716	. . . . . . . . . 10 ⊢ (𝐷 ∈ (∞Met‘𝑋) → dom 𝐷 = (𝑋 × 𝑋))
7	6	dmeqd 5894	. . . . . . . . 9 ⊢ (𝐷 ∈ (∞Met‘𝑋) → dom dom 𝐷 = dom (𝑋 × 𝑋))
8		dmxpid 5919	. . . . . . . . 9 ⊢ dom (𝑋 × 𝑋) = 𝑋
9	7, 8	eqtrdi 2813	. . . . . . . 8 ⊢ (𝐷 ∈ (∞Met‘𝑋) → dom dom 𝐷 = 𝑋)
10	4, 9	sylan9eqr 2819	. . . . . . 7 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → dom dom 𝑑 = 𝑋)
11	10	opeq2d 4844	. . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → ⟨(Base‘ndx), dom dom 𝑑⟩ = ⟨(Base‘ndx), 𝑋⟩)
12		simpr 489	. . . . . . 7 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → 𝑑 = 𝐷)
13	12	opeq2d 4844	. . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → ⟨(dist‘ndx), 𝑑⟩ = ⟨(dist‘ndx), 𝐷⟩)
14	11, 13	preq12d 4706	. . . . 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⟩}
16	14, 15	eqtr4di 2815	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → {⟨(Base‘ndx), dom dom 𝑑⟩, ⟨(dist‘ndx), 𝑑⟩} = 𝑀)
17	12	fveq2d 6885	. . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → (MetOpen‘𝑑) = (MetOpen‘𝐷))
18	17	opeq2d 4844	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → ⟨(TopSet‘ndx), (MetOpen‘𝑑)⟩ = ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩)
19	16, 18	oveq12d 7430	. . 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 6912	. . . 4 ⊢ (∞Met‘𝑋) ⊆ ∪ ran ∞Met
21	20	sseli 3932	. . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 ∈ ∪ ran ∞Met)
22		ovexd 7447	. . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩) ∈ V)
23	2, 19, 21, 22	fvmptd2 6998	. 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (toMetSp‘𝐷) = (𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩))
24	1, 23	eqtrid 2809	1 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐾 = (𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩))

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   = wceq 1569   ∈ wcel 2142  Vcvv 3454  {cpr 4590  ⟨cop 4594  ∪ cuni 4871   × cxp 5658  dom cdm 5660  ran crn 5661  ‘cfv 6536  (class class class)co 7412  ℝ^*cxr 11248   sSet csts 17229  ndxcnx 17259  Basecbs 17275  TopSetcts 17322  distcds 17325  ∞Metcxmet 21518  MetOpencmopn 21523  toMetSpctms 24487

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734  ax-cnex 11162  ax-resscn 11163

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7415  df-oprab 7416  df-mpo 7417  df-map 8824  df-xr 11253  df-xmet 21526  df-tms 24490

This theorem is used by:  tmslem  24650