Theorem blfvalps 22596

Description: The value of the ball function. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) (Revised by Thierry Arnoux, 11-Feb-2018.)

Assertion

Ref	Expression
blfvalps	⊢ (𝐷 ∈ (PsMet‘𝑋) → (ball‘𝐷) = (𝑥 ∈ 𝑋, 𝑟 ∈ ℝ* ↦ {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}))

Distinct variable groups:   𝑥,𝑟,𝑦,𝐷   𝑋,𝑟,𝑥,𝑦

Proof of Theorem blfvalps

Dummy variable 𝑑 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-bl 20137	. . 3 ⊢ ball = (𝑑 ∈ V ↦ (𝑥 ∈ dom dom 𝑑, 𝑟 ∈ ℝ* ↦ {𝑦 ∈ dom dom 𝑑 ∣ (𝑥𝑑𝑦) < 𝑟}))
2	1	a1i 11	. 2 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ball = (𝑑 ∈ V ↦ (𝑥 ∈ dom dom 𝑑, 𝑟 ∈ ℝ* ↦ {𝑦 ∈ dom dom 𝑑 ∣ (𝑥𝑑𝑦) < 𝑟})))
3		dmeq 5569	. . . . 5 ⊢ (𝑑 = 𝐷 → dom 𝑑 = dom 𝐷)
4	3	dmeqd 5571	. . . 4 ⊢ (𝑑 = 𝐷 → dom dom 𝑑 = dom dom 𝐷)
5		psmetdmdm 22518	. . . . 5 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝑋 = dom dom 𝐷)
6	5	eqcomd 2784	. . . 4 ⊢ (𝐷 ∈ (PsMet‘𝑋) → dom dom 𝐷 = 𝑋)
7	4, 6	sylan9eqr 2836	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → dom dom 𝑑 = 𝑋)
8		eqidd 2779	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → ℝ* = ℝ*)
9		simpr 479	. . . . . 6 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → 𝑑 = 𝐷)
10	9	oveqd 6939	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → (𝑥𝑑𝑦) = (𝑥𝐷𝑦))
11	10	breq1d 4896	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → ((𝑥𝑑𝑦) < 𝑟 ↔ (𝑥𝐷𝑦) < 𝑟))
12	7, 11	rabeqbidv 3392	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → {𝑦 ∈ dom dom 𝑑 ∣ (𝑥𝑑𝑦) < 𝑟} = {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟})
13	7, 8, 12	mpt2eq123dv 6994	. 2 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → (𝑥 ∈ dom dom 𝑑, 𝑟 ∈ ℝ* ↦ {𝑦 ∈ dom dom 𝑑 ∣ (𝑥𝑑𝑦) < 𝑟}) = (𝑥 ∈ 𝑋, 𝑟 ∈ ℝ* ↦ {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}))
14		elex 3414	. 2 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷 ∈ V)
15		ssrab2 3908	. . . . . 6 ⊢ {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ⊆ 𝑋
16		elfvdm 6478	. . . . . . . 8 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ dom PsMet)
17	16	adantr 474	. . . . . . 7 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ*)) → 𝑋 ∈ dom PsMet)
18		elpw2g 5061	. . . . . . 7 ⊢ (𝑋 ∈ dom PsMet → ({𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ∈ 𝒫 𝑋 ↔ {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ⊆ 𝑋))
19	17, 18	syl 17	. . . . . 6 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ*)) → ({𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ∈ 𝒫 𝑋 ↔ {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ⊆ 𝑋))
20	15, 19	mpbiri 250	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ*)) → {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ∈ 𝒫 𝑋)
21	20	ralrimivva 3153	. . . 4 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ∀𝑥 ∈ 𝑋 ∀𝑟 ∈ ℝ* {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ∈ 𝒫 𝑋)
22		eqid 2778	. . . . 5 ⊢ (𝑥 ∈ 𝑋, 𝑟 ∈ ℝ* ↦ {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}) = (𝑥 ∈ 𝑋, 𝑟 ∈ ℝ* ↦ {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟})
23	22	fmpt2 7517	. . . 4 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑟 ∈ ℝ* {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ∈ 𝒫 𝑋 ↔ (𝑥 ∈ 𝑋, 𝑟 ∈ ℝ* ↦ {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}):(𝑋 × ℝ*)⟶𝒫 𝑋)
24	21, 23	sylib 210	. . 3 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝑥 ∈ 𝑋, 𝑟 ∈ ℝ* ↦ {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}):(𝑋 × ℝ*)⟶𝒫 𝑋)
25		xrex 12134	. . . 4 ⊢ ℝ* ∈ V
26		xpexg 7237	. . . 4 ⊢ ((𝑋 ∈ dom PsMet ∧ ℝ* ∈ V) → (𝑋 × ℝ*) ∈ V)
27	16, 25, 26	sylancl 580	. . 3 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝑋 × ℝ*) ∈ V)
28	16	pwexd 5091	. . 3 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝒫 𝑋 ∈ V)
29		fex2 7400	. . 3 ⊢ (((𝑥 ∈ 𝑋, 𝑟 ∈ ℝ* ↦ {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}):(𝑋 × ℝ*)⟶𝒫 𝑋 ∧ (𝑋 × ℝ*) ∈ V ∧ 𝒫 𝑋 ∈ V) → (𝑥 ∈ 𝑋, 𝑟 ∈ ℝ* ↦ {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}) ∈ V)
30	24, 27, 28, 29	syl3anc 1439	. 2 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝑥 ∈ 𝑋, 𝑟 ∈ ℝ* ↦ {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}) ∈ V)
31	2, 13, 14, 30	fvmptd 6548	1 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (ball‘𝐷) = (𝑥 ∈ 𝑋, 𝑟 ∈ ℝ* ↦ {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1601   ∈ wcel 2107  ∀wral 3090  {crab 3094  Vcvv 3398   ⊆ wss 3792  𝒫 cpw 4379   class class class wbr 4886   ↦ cmpt 4965   × cxp 5353  dom cdm 5355  ⟶wf 6131  ‘cfv 6135  (class class class)co 6922   ↦ cmpt2 6924  ℝ^*cxr 10410   < clt 10411  PsMetcpsmet 20126  ballcbl 20129

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-cnex 10328  ax-resscn 10329

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-1st 7445  df-2nd 7446  df-map 8142  df-xr 10415  df-psmet 20134  df-bl 20137

This theorem is referenced by:  blfval  22597  blvalps  22598  blfps  22619