Theorem blfvalps 22996

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 20543	. 2 ⊢ ball = (𝑑 ∈ V ↦ (𝑥 ∈ dom dom 𝑑, 𝑟 ∈ ℝ* ↦ {𝑦 ∈ dom dom 𝑑 ∣ (𝑥𝑑𝑦) < 𝑟}))
2		dmeq 5775	. . . . 5 ⊢ (𝑑 = 𝐷 → dom 𝑑 = dom 𝐷)
3	2	dmeqd 5777	. . . 4 ⊢ (𝑑 = 𝐷 → dom dom 𝑑 = dom dom 𝐷)
4		psmetdmdm 22918	. . . . 5 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝑋 = dom dom 𝐷)
5	4	eqcomd 2830	. . . 4 ⊢ (𝐷 ∈ (PsMet‘𝑋) → dom dom 𝐷 = 𝑋)
6	3, 5	sylan9eqr 2881	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → dom dom 𝑑 = 𝑋)
7		eqidd 2825	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → ℝ* = ℝ*)
8		simpr 487	. . . . . 6 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → 𝑑 = 𝐷)
9	8	oveqd 7176	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → (𝑥𝑑𝑦) = (𝑥𝐷𝑦))
10	9	breq1d 5079	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → ((𝑥𝑑𝑦) < 𝑟 ↔ (𝑥𝐷𝑦) < 𝑟))
11	6, 10	rabeqbidv 3488	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → {𝑦 ∈ dom dom 𝑑 ∣ (𝑥𝑑𝑦) < 𝑟} = {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟})
12	6, 7, 11	mpoeq123dv 7232	. 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}))
13		elex 3515	. 2 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷 ∈ V)
14		ssrab2 4059	. . . . . 6 ⊢ {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ⊆ 𝑋
15		elfvdm 6705	. . . . . . . 8 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ dom PsMet)
16	15	adantr 483	. . . . . . 7 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ*)) → 𝑋 ∈ dom PsMet)
17		elpw2g 5250	. . . . . . 7 ⊢ (𝑋 ∈ dom PsMet → ({𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ∈ 𝒫 𝑋 ↔ {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ⊆ 𝑋))
18	16, 17	syl 17	. . . . . 6 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ*)) → ({𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ∈ 𝒫 𝑋 ↔ {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ⊆ 𝑋))
19	14, 18	mpbiri 260	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ*)) → {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ∈ 𝒫 𝑋)
20	19	ralrimivva 3194	. . . 4 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ∀𝑥 ∈ 𝑋 ∀𝑟 ∈ ℝ* {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ∈ 𝒫 𝑋)
21		eqid 2824	. . . . 5 ⊢ (𝑥 ∈ 𝑋, 𝑟 ∈ ℝ* ↦ {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}) = (𝑥 ∈ 𝑋, 𝑟 ∈ ℝ* ↦ {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟})
22	21	fmpo 7769	. . . 4 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑟 ∈ ℝ* {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ∈ 𝒫 𝑋 ↔ (𝑥 ∈ 𝑋, 𝑟 ∈ ℝ* ↦ {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}):(𝑋 × ℝ*)⟶𝒫 𝑋)
23	20, 22	sylib 220	. . 3 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝑥 ∈ 𝑋, 𝑟 ∈ ℝ* ↦ {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}):(𝑋 × ℝ*)⟶𝒫 𝑋)
24		xrex 12389	. . . 4 ⊢ ℝ* ∈ V
25		xpexg 7476	. . . 4 ⊢ ((𝑋 ∈ dom PsMet ∧ ℝ* ∈ V) → (𝑋 × ℝ*) ∈ V)
26	15, 24, 25	sylancl 588	. . 3 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝑋 × ℝ*) ∈ V)
27	15	pwexd 5283	. . 3 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝒫 𝑋 ∈ V)
28		fex2 7641	. . 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)
29	23, 26, 27, 28	syl3anc 1367	. 2 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝑥 ∈ 𝑋, 𝑟 ∈ ℝ* ↦ {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}) ∈ V)
30	1, 12, 13, 29	fvmptd2 6779	1 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (ball‘𝐷) = (𝑥 ∈ 𝑋, 𝑟 ∈ ℝ* ↦ {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1536   ∈ wcel 2113  ∀wral 3141  {crab 3145  Vcvv 3497   ⊆ wss 3939  𝒫 cpw 4542   class class class wbr 5069   × cxp 5556  dom cdm 5558  ⟶wf 6354  ‘cfv 6358  (class class class)co 7159   ∈ cmpo 7161  ℝ^*cxr 10677   < clt 10678  PsMetcpsmet 20532  ballcbl 20535

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464  ax-cnex 10596  ax-resscn 10597

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-fv 6366  df-ov 7162  df-oprab 7163  df-mpo 7164  df-1st 7692  df-2nd 7693  df-map 8411  df-xr 10682  df-psmet 20540  df-bl 20543

This theorem is referenced by:  blfval  22997  blvalps  22998  blfps  23019