Theorem blfvalps 15075

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 14526	. . 3 ⊢ ball = (𝑑 ∈ V ↦ (𝑥 ∈ dom dom 𝑑, 𝑟 ∈ ℝ* ↦ {𝑦 ∈ dom dom 𝑑 ∣ (𝑥𝑑𝑦) < 𝑟}))
2	1	a1i 9	. 2 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ball = (𝑑 ∈ V ↦ (𝑥 ∈ dom dom 𝑑, 𝑟 ∈ ℝ* ↦ {𝑦 ∈ dom dom 𝑑 ∣ (𝑥𝑑𝑦) < 𝑟})))
3		dmeq 4923	. . . . 5 ⊢ (𝑑 = 𝐷 → dom 𝑑 = dom 𝐷)
4	3	dmeqd 4925	. . . 4 ⊢ (𝑑 = 𝐷 → dom dom 𝑑 = dom dom 𝐷)
5		psmetdmdm 15014	. . . . 5 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝑋 = dom dom 𝐷)
6	5	eqcomd 2235	. . . 4 ⊢ (𝐷 ∈ (PsMet‘𝑋) → dom dom 𝐷 = 𝑋)
7	4, 6	sylan9eqr 2284	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → dom dom 𝑑 = 𝑋)
8		eqidd 2230	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → ℝ* = ℝ*)
9		simpr 110	. . . . . 6 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → 𝑑 = 𝐷)
10	9	oveqd 6024	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → (𝑥𝑑𝑦) = (𝑥𝐷𝑦))
11	10	breq1d 4093	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → ((𝑥𝑑𝑦) < 𝑟 ↔ (𝑥𝐷𝑦) < 𝑟))
12	7, 11	rabeqbidv 2794	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → {𝑦 ∈ dom dom 𝑑 ∣ (𝑥𝑑𝑦) < 𝑟} = {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟})
13	7, 8, 12	mpoeq123dv 6072	. 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 2811	. 2 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷 ∈ V)
15		ssrab2 3309	. . . . . 6 ⊢ {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ⊆ 𝑋
16		psmetrel 15012	. . . . . . . . 9 ⊢ Rel PsMet
17		relelfvdm 5661	. . . . . . . . 9 ⊢ ((Rel PsMet ∧ 𝐷 ∈ (PsMet‘𝑋)) → 𝑋 ∈ dom PsMet)
18	16, 17	mpan 424	. . . . . . . 8 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ dom PsMet)
19	18	adantr 276	. . . . . . 7 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ*)) → 𝑋 ∈ dom PsMet)
20		elpw2g 4240	. . . . . . 7 ⊢ (𝑋 ∈ dom PsMet → ({𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ∈ 𝒫 𝑋 ↔ {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ⊆ 𝑋))
21	19, 20	syl 14	. . . . . 6 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ*)) → ({𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ∈ 𝒫 𝑋 ↔ {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ⊆ 𝑋))
22	15, 21	mpbiri 168	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ*)) → {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ∈ 𝒫 𝑋)
23	22	ralrimivva 2612	. . . 4 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ∀𝑥 ∈ 𝑋 ∀𝑟 ∈ ℝ* {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ∈ 𝒫 𝑋)
24		eqid 2229	. . . . 5 ⊢ (𝑥 ∈ 𝑋, 𝑟 ∈ ℝ* ↦ {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}) = (𝑥 ∈ 𝑋, 𝑟 ∈ ℝ* ↦ {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟})
25	24	fmpo 6353	. . . 4 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑟 ∈ ℝ* {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ∈ 𝒫 𝑋 ↔ (𝑥 ∈ 𝑋, 𝑟 ∈ ℝ* ↦ {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}):(𝑋 × ℝ*)⟶𝒫 𝑋)
26	23, 25	sylib 122	. . 3 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝑥 ∈ 𝑋, 𝑟 ∈ ℝ* ↦ {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}):(𝑋 × ℝ*)⟶𝒫 𝑋)
27		xrex 10064	. . . 4 ⊢ ℝ* ∈ V
28		xpexg 4833	. . . 4 ⊢ ((𝑋 ∈ dom PsMet ∧ ℝ* ∈ V) → (𝑋 × ℝ*) ∈ V)
29	18, 27, 28	sylancl 413	. . 3 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝑋 × ℝ*) ∈ V)
30	18	pwexd 4265	. . 3 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝒫 𝑋 ∈ V)
31		fex2 5494	. . 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)
32	26, 29, 30, 31	syl3anc 1271	. 2 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝑥 ∈ 𝑋, 𝑟 ∈ ℝ* ↦ {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}) ∈ V)
33	2, 13, 14, 32	fvmptd 5717	1 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (ball‘𝐷) = (𝑥 ∈ 𝑋, 𝑟 ∈ ℝ* ↦ {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1395   ∈ wcel 2200  ∀wral 2508  {crab 2512  Vcvv 2799   ⊆ wss 3197  𝒫 cpw 3649   class class class wbr 4083   ↦ cmpt 4145   × cxp 4717  dom cdm 4719  Rel wrel 4724  ⟶wf 5314  ‘cfv 5318  (class class class)co 6007   ∈ cmpo 6009  ℝ^*cxr 8191   < clt 8192  PsMetcpsmet 14515  ballcbl 14518

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8101  ax-resscn 8102

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fv 5326  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-map 6805  df-pnf 8194  df-mnf 8195  df-xr 8196  df-psmet 14523  df-bl 14526

This theorem is referenced by:  blfval  15076  blvalps  15078  blfps  15099