Theorem metuval 24057

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 variable 𝑑 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-metu 20942	. 2 ⊢ metUnif = (𝑑 ∈ ∪ ran PsMet ↦ ((dom dom 𝑑 × dom dom 𝑑)filGenran (𝑎 ∈ ℝ+ ↦ (◡𝑑 “ (0[,)𝑎)))))
2		simpr 485	. . . . . . 7 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → 𝑑 = 𝐷)
3	2	dmeqd 5905	. . . . . 6 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → dom 𝑑 = dom 𝐷)
4	3	dmeqd 5905	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → dom dom 𝑑 = dom dom 𝐷)
5		psmetdmdm 23810	. . . . . 6 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝑋 = dom dom 𝐷)
6	5	adantr 481	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → 𝑋 = dom dom 𝐷)
7	4, 6	eqtr4d 2775	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → dom dom 𝑑 = 𝑋)
8	7	sqxpeqd 5708	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → (dom dom 𝑑 × dom dom 𝑑) = (𝑋 × 𝑋))
9		simplr 767	. . . . . . 7 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) ∧ 𝑎 ∈ ℝ+) → 𝑑 = 𝐷)
10	9	cnveqd 5875	. . . . . 6 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) ∧ 𝑎 ∈ ℝ+) → ◡𝑑 = ◡𝐷)
11	10	imaeq1d 6058	. . . . 5 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) ∧ 𝑎 ∈ ℝ+) → (◡𝑑 “ (0[,)𝑎)) = (◡𝐷 “ (0[,)𝑎)))
12	11	mpteq2dva 5248	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → (𝑎 ∈ ℝ+ ↦ (◡𝑑 “ (0[,)𝑎))) = (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))))
13	12	rneqd 5937	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → ran (𝑎 ∈ ℝ+ ↦ (◡𝑑 “ (0[,)𝑎))) = ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))))
14	8, 13	oveq12d 7426	. 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		elfvunirn 6923	. 2 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷 ∈ ∪ ran PsMet)
16		ovexd 7443	. 2 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ((𝑋 × 𝑋)filGenran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))) ∈ V)
17	1, 14, 15, 16	fvmptd2 7006	1 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (metUnif‘𝐷) = ((𝑋 × 𝑋)filGenran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  Vcvv 3474  ∪ cuni 4908   ↦ cmpt 5231   × cxp 5674  ^◡ccnv 5675  dom cdm 5676  ran crn 5677   “ cima 5679  ‘cfv 6543  (class class class)co 7408  0cc0 11109  ℝ⁺crp 12973  [,)cico 13325  PsMetcpsmet 20927  filGencfg 20932  metUnifcmetu 20934

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-cnex 11165  ax-resscn 11166

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-map 8821  df-xr 11251  df-psmet 20935  df-metu 20942

This theorem is referenced by:  metuust  24068  cfilucfil2  24069  metuel  24072  psmetutop  24075  restmetu  24078  metucn  24079