Theorem metuval 23611

Description: Value of the uniform structure generated by metric 𝐷. (Contributed by Thierry Arnoux, 1-Dec-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)

Assertion

Ref	Expression
metuval	⊢ (𝐷 ∈ (PsMet‘𝑋) → (metUnif‘𝐷) = ((𝑋 × 𝑋)filGenran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))))

Distinct variable groups:   𝐷,𝑎   𝑋,𝑎

Proof of Theorem metuval

Dummy variables 𝑢 𝑑 𝑣 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-metu 20509	. 2 ⊢ metUnif = (𝑑 ∈ ∪ ran PsMet ↦ ((dom dom 𝑑 × dom dom 𝑑)filGenran (𝑎 ∈ ℝ+ ↦ (◡𝑑 “ (0[,)𝑎)))))
2		simpr 484	. . . . . . 7 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → 𝑑 = 𝐷)
3	2	dmeqd 5803	. . . . . 6 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → dom 𝑑 = dom 𝐷)
4	3	dmeqd 5803	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → dom dom 𝑑 = dom dom 𝐷)
5		psmetdmdm 23366	. . . . . 6 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝑋 = dom dom 𝐷)
6	5	adantr 480	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → 𝑋 = dom dom 𝐷)
7	4, 6	eqtr4d 2781	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → dom dom 𝑑 = 𝑋)
8	7	sqxpeqd 5612	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → (dom dom 𝑑 × dom dom 𝑑) = (𝑋 × 𝑋))
9		simplr 765	. . . . . . 7 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) ∧ 𝑎 ∈ ℝ+) → 𝑑 = 𝐷)
10	9	cnveqd 5773	. . . . . 6 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) ∧ 𝑎 ∈ ℝ+) → ◡𝑑 = ◡𝐷)
11	10	imaeq1d 5957	. . . . 5 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) ∧ 𝑎 ∈ ℝ+) → (◡𝑑 “ (0[,)𝑎)) = (◡𝐷 “ (0[,)𝑎)))
12	11	mpteq2dva 5170	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → (𝑎 ∈ ℝ+ ↦ (◡𝑑 “ (0[,)𝑎))) = (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))))
13	12	rneqd 5836	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → ran (𝑎 ∈ ℝ+ ↦ (◡𝑑 “ (0[,)𝑎))) = ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))))
14	8, 13	oveq12d 7273	. 2 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → ((dom dom 𝑑 × dom dom 𝑑)filGenran (𝑎 ∈ ℝ+ ↦ (◡𝑑 “ (0[,)𝑎)))) = ((𝑋 × 𝑋)filGenran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))))
15		elfvdm 6788	. . . 4 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ dom PsMet)
16		fveq2 6756	. . . . . 6 ⊢ (𝑥 = 𝑋 → (PsMet‘𝑥) = (PsMet‘𝑋))
17	16	eleq2d 2824	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝐷 ∈ (PsMet‘𝑥) ↔ 𝐷 ∈ (PsMet‘𝑋)))
18	17	rspcev 3552	. . . 4 ⊢ ((𝑋 ∈ dom PsMet ∧ 𝐷 ∈ (PsMet‘𝑋)) → ∃𝑥 ∈ dom PsMet𝐷 ∈ (PsMet‘𝑥))
19	15, 18	mpancom 684	. . 3 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ∃𝑥 ∈ dom PsMet𝐷 ∈ (PsMet‘𝑥))
20		df-psmet 20502	. . . . 5 ⊢ PsMet = (𝑦 ∈ V ↦ {𝑢 ∈ (ℝ* ↑m (𝑦 × 𝑦)) ∣ ∀𝑧 ∈ 𝑦 ((𝑧𝑢𝑧) = 0 ∧ ∀𝑤 ∈ 𝑦 ∀𝑣 ∈ 𝑦 (𝑧𝑢𝑤) ≤ ((𝑣𝑢𝑧) +𝑒 (𝑣𝑢𝑤)))})
21	20	funmpt2 6457	. . . 4 ⊢ Fun PsMet
22		elunirn 7106	. . . 4 ⊢ (Fun PsMet → (𝐷 ∈ ∪ ran PsMet ↔ ∃𝑥 ∈ dom PsMet𝐷 ∈ (PsMet‘𝑥)))
23	21, 22	ax-mp 5	. . 3 ⊢ (𝐷 ∈ ∪ ran PsMet ↔ ∃𝑥 ∈ dom PsMet𝐷 ∈ (PsMet‘𝑥))
24	19, 23	sylibr 233	. 2 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷 ∈ ∪ ran PsMet)
25		ovexd 7290	. 2 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ((𝑋 × 𝑋)filGenran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))) ∈ V)
26	1, 14, 24, 25	fvmptd2 6865	1 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (metUnif‘𝐷) = ((𝑋 × 𝑋)filGenran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064  {crab 3067  Vcvv 3422  ∪ cuni 4836   class class class wbr 5070   ↦ cmpt 5153   × cxp 5578  ^◡ccnv 5579  dom cdm 5580  ran crn 5581   “ cima 5583  Fun wfun 6412  ‘cfv 6418  (class class class)co 7255   ↑_m cmap 8573  0cc0 10802  ℝ^*cxr 10939   ≤ cle 10941  ℝ⁺crp 12659   +_𝑒 cxad 12775  [,)cico 13010  PsMetcpsmet 20494  filGencfg 20499  metUnifcmetu 20501

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-map 8575  df-xr 10944  df-psmet 20502  df-metu 20509

This theorem is referenced by:  metuust  23622  cfilucfil2  23623  metuel  23626  psmetutop  23629  restmetu  23632  metucn  23633