Theorem blfps 24416

Description: Mapping of a ball. (Contributed by NM, 7-May-2007.) (Revised by Mario Carneiro, 23-Aug-2015.) (Revised by Thierry Arnoux, 11-Mar-2018.)

Assertion

Ref	Expression
blfps	⊢ (𝐷 ∈ (PsMet‘𝑋) → (ball‘𝐷):(𝑋 × ℝ*)⟶𝒫 𝑋)

Proof of Theorem blfps

Dummy variables 𝑥 𝑟 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssrab2 4080	. . . . . 6 ⊢ {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ⊆ 𝑋
2		elfvdm 6943	. . . . . . 7 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ dom PsMet)
3		elpw2g 5333	. . . . . . 7 ⊢ (𝑋 ∈ dom PsMet → ({𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ∈ 𝒫 𝑋 ↔ {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ⊆ 𝑋))
4	2, 3	syl 17	. . . . . 6 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ({𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ∈ 𝒫 𝑋 ↔ {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ⊆ 𝑋))
5	1, 4	mpbiri 258	. . . . 5 ⊢ (𝐷 ∈ (PsMet‘𝑋) → {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ∈ 𝒫 𝑋)
6	5	a1d 25	. . . 4 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ*) → {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ∈ 𝒫 𝑋))
7	6	ralrimivv 3200	. . 3 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ∀𝑥 ∈ 𝑋 ∀𝑟 ∈ ℝ* {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ∈ 𝒫 𝑋)
8		eqid 2737	. . . 4 ⊢ (𝑥 ∈ 𝑋, 𝑟 ∈ ℝ* ↦ {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}) = (𝑥 ∈ 𝑋, 𝑟 ∈ ℝ* ↦ {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟})
9	8	fmpo 8093	. . 3 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑟 ∈ ℝ* {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ∈ 𝒫 𝑋 ↔ (𝑥 ∈ 𝑋, 𝑟 ∈ ℝ* ↦ {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}):(𝑋 × ℝ*)⟶𝒫 𝑋)
10	7, 9	sylib 218	. 2 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝑥 ∈ 𝑋, 𝑟 ∈ ℝ* ↦ {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}):(𝑋 × ℝ*)⟶𝒫 𝑋)
11		blfvalps 24393	. . 3 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (ball‘𝐷) = (𝑥 ∈ 𝑋, 𝑟 ∈ ℝ* ↦ {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}))
12	11	feq1d 6720	. 2 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ((ball‘𝐷):(𝑋 × ℝ*)⟶𝒫 𝑋 ↔ (𝑥 ∈ 𝑋, 𝑟 ∈ ℝ* ↦ {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}):(𝑋 × ℝ*)⟶𝒫 𝑋))
13	10, 12	mpbird 257	1 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (ball‘𝐷):(𝑋 × ℝ*)⟶𝒫 𝑋)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2108  ∀wral 3061  {crab 3436   ⊆ wss 3951  𝒫 cpw 4600   class class class wbr 5143   × cxp 5683  dom cdm 5685  ⟶wf 6557  ‘cfv 6561  (class class class)co 7431   ∈ cmpo 7433  ℝ^*cxr 11294   < clt 11295  PsMetcpsmet 21348  ballcbl 21351

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 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-map 8868  df-xr 11299  df-psmet 21356  df-bl 21359

This theorem is referenced by:  blrnps  24418  blelrnps  24426  unirnblps  24429