Theorem tmsval 24376

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 24217	. . 3 ⊢ toMetSp = (𝑑 ∈ ∪ ran ∞Met ↦ ({⟨(Base‘ndx), dom dom 𝑑⟩, ⟨(dist‘ndx), 𝑑⟩} sSet ⟨(TopSet‘ndx), (MetOpen‘𝑑)⟩))
3		dmeq 5870	. . . . . . . . 9 ⊢ (𝑑 = 𝐷 → dom 𝑑 = dom 𝐷)
4	3	dmeqd 5872	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → dom dom 𝑑 = dom dom 𝐷)
5		xmetf 24224	. . . . . . . . . . 11 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
6	5	fdmd 6701	. . . . . . . . . 10 ⊢ (𝐷 ∈ (∞Met‘𝑋) → dom 𝐷 = (𝑋 × 𝑋))
7	6	dmeqd 5872	. . . . . . . . 9 ⊢ (𝐷 ∈ (∞Met‘𝑋) → dom dom 𝐷 = dom (𝑋 × 𝑋))
8		dmxpid 5897	. . . . . . . . 9 ⊢ dom (𝑋 × 𝑋) = 𝑋
9	7, 8	eqtrdi 2781	. . . . . . . 8 ⊢ (𝐷 ∈ (∞Met‘𝑋) → dom dom 𝐷 = 𝑋)
10	4, 9	sylan9eqr 2787	. . . . . . 7 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → dom dom 𝑑 = 𝑋)
11	10	opeq2d 4847	. . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → ⟨(Base‘ndx), dom dom 𝑑⟩ = ⟨(Base‘ndx), 𝑋⟩)
12		simpr 484	. . . . . . 7 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → 𝑑 = 𝐷)
13	12	opeq2d 4847	. . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → ⟨(dist‘ndx), 𝑑⟩ = ⟨(dist‘ndx), 𝐷⟩)
14	11, 13	preq12d 4708	. . . . 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 2783	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → {⟨(Base‘ndx), dom dom 𝑑⟩, ⟨(dist‘ndx), 𝑑⟩} = 𝑀)
17	12	fveq2d 6865	. . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → (MetOpen‘𝑑) = (MetOpen‘𝐷))
18	17	opeq2d 4847	. . . 4 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑑 = 𝐷) → ⟨(TopSet‘ndx), (MetOpen‘𝑑)⟩ = ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩)
19	16, 18	oveq12d 7408	. . 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 6894	. . . 4 ⊢ (∞Met‘𝑋) ⊆ ∪ ran ∞Met
21	20	sseli 3945	. . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 ∈ ∪ ran ∞Met)
22		ovexd 7425	. . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩) ∈ V)
23	2, 19, 21, 22	fvmptd2 6979	. 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (toMetSp‘𝐷) = (𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩))
24	1, 23	eqtrid 2777	1 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐾 = (𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3450  {cpr 4594  ⟨cop 4598  ∪ cuni 4874   × cxp 5639  dom cdm 5641  ran crn 5642  ‘cfv 6514  (class class class)co 7390  ℝ^*cxr 11214   sSet csts 17140  ndxcnx 17170  Basecbs 17186  TopSetcts 17233  distcds 17236  ∞Metcxmet 21256  MetOpencmopn 21261  toMetSpctms 24214

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-map 8804  df-xr 11219  df-xmet 21264  df-tms 24217

This theorem is referenced by:  tmslem  24377