Theorem tmsval 22286

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 22127	. . . 4 ⊢ toMetSp = (𝑑 ∈ ∪ ran ∞Met ↦ ({⟨(Base‘ndx), dom dom 𝑑⟩, ⟨(dist‘ndx), 𝑑⟩} sSet ⟨(TopSet‘ndx), (MetOpen‘𝑑)⟩))
3	2	a1i 11	. . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → toMetSp = (𝑑 ∈ ∪ ran ∞Met ↦ ({⟨(Base‘ndx), dom dom 𝑑⟩, ⟨(dist‘ndx), 𝑑⟩} sSet ⟨(TopSet‘ndx), (MetOpen‘𝑑)⟩)))
4		dmeq 5324	. . . . . . . . 9 ⊢ (𝑑 = 𝐷 → dom 𝑑 = dom 𝐷)
5	4	dmeqd 5326	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → dom dom 𝑑 = dom dom 𝐷)
6		xmetf 22134	. . . . . . . . . . 11 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
7		fdm 6051	. . . . . . . . . . 11 ⊢ (𝐷:(𝑋 × 𝑋)⟶ℝ* → dom 𝐷 = (𝑋 × 𝑋))
8	6, 7	syl 17	. . . . . . . . . 10 ⊢ (𝐷 ∈ (∞Met‘𝑋) → dom 𝐷 = (𝑋 × 𝑋))
9	8	dmeqd 5326	. . . . . . . . 9 ⊢ (𝐷 ∈ (∞Met‘𝑋) → dom dom 𝐷 = dom (𝑋 × 𝑋))
10		dmxpid 5345	. . . . . . . . 9 ⊢ dom (𝑋 × 𝑋) = 𝑋
11	9, 10	syl6eq 2672	. . . . . . . 8 ⊢ (𝐷 ∈ (∞Met‘𝑋) → dom dom 𝐷 = 𝑋)
12	5, 11	sylan9eqr 2678	. . . . . . 7 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → dom dom 𝑑 = 𝑋)
13	12	opeq2d 4409	. . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → ⟨(Base‘ndx), dom dom 𝑑⟩ = ⟨(Base‘ndx), 𝑋⟩)
14		simpr 477	. . . . . . 7 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → 𝑑 = 𝐷)
15	14	opeq2d 4409	. . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → ⟨(dist‘ndx), 𝑑⟩ = ⟨(dist‘ndx), 𝐷⟩)
16	13, 15	preq12d 4276	. . . . 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⟩}
18	16, 17	syl6eqr 2674	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → {⟨(Base‘ndx), dom dom 𝑑⟩, ⟨(dist‘ndx), 𝑑⟩} = 𝑀)
19	14	fveq2d 6195	. . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → (MetOpen‘𝑑) = (MetOpen‘𝐷))
20	19	opeq2d 4409	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → ⟨(TopSet‘ndx), (MetOpen‘𝑑)⟩ = ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩)
21	18, 20	oveq12d 6668	. . 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 6217	. . . 4 ⊢ (∞Met‘𝑋) ⊆ ∪ ran ∞Met
23	22	sseli 3599	. . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 ∈ ∪ ran ∞Met)
24		ovexd 6680	. . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩) ∈ V)
25	3, 21, 23, 24	fvmptd 6288	. 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (toMetSp‘𝐷) = (𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩))
26	1, 25	syl5eq 2668	1 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐾 = (𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Vcvv 3200  {cpr 4179  ⟨cop 4183  ∪ cuni 4436   ↦ cmpt 4729   × cxp 5112  dom cdm 5114  ran crn 5115  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  ℝ^*cxr 10073  ndxcnx 15854   sSet csts 15855  Basecbs 15857  TopSetcts 15947  distcds 15950  ∞Metcxmt 19731  MetOpencmopn 19736  toMetSpctmt 22124

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-xr 10078  df-xmet 19739  df-tms 22127

This theorem is referenced by:  tmslem  22287