Theorem blfvalps 24431

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 21407	. 2 ⊢ ball = (𝑑 ∈ V ↦ (𝑥 ∈ dom dom 𝑑, 𝑟 ∈ ℝ* ↦ {𝑦 ∈ dom dom 𝑑 ∣ (𝑥𝑑𝑦) < 𝑟}))
2		dmeq 5875	. . . . 5 ⊢ (𝑑 = 𝐷 → dom 𝑑 = dom 𝐷)
3	2	dmeqd 5877	. . . 4 ⊢ (𝑑 = 𝐷 → dom dom 𝑑 = dom dom 𝐷)
4		psmetdmdm 24353	. . . . 5 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝑋 = dom dom 𝐷)
5	4	eqcomd 2767	. . . 4 ⊢ (𝐷 ∈ (PsMet‘𝑋) → dom dom 𝐷 = 𝑋)
6	3, 5	sylan9eqr 2818	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → dom dom 𝑑 = 𝑋)
7		eqidd 2762	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → ℝ* = ℝ*)
8		simpr 488	. . . . . 6 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → 𝑑 = 𝐷)
9	8	oveqd 7408	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → (𝑥𝑑𝑦) = (𝑥𝐷𝑦))
10	9	breq1d 5107	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → ((𝑥𝑑𝑦) < 𝑟 ↔ (𝑥𝐷𝑦) < 𝑟))
11	6, 10	rabeqbidv 3431	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → {𝑦 ∈ dom dom 𝑑 ∣ (𝑥𝑑𝑦) < 𝑟} = {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟})
12	6, 7, 11	mpoeq123dv 7466	. 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 3474	. 2 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷 ∈ V)
14		ssrab2 4031	. . . . . 6 ⊢ {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ⊆ 𝑋
15		elfvdm 6896	. . . . . . . 8 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ dom PsMet)
16	15	adantr 484	. . . . . . 7 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ*)) → 𝑋 ∈ dom PsMet)
17		elpw2g 5286	. . . . . . 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 3204	. . . 4 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ∀𝑥 ∈ 𝑋 ∀𝑟 ∈ ℝ* {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟} ∈ 𝒫 𝑋)
21		eqid 2761	. . . . 5 ⊢ (𝑥 ∈ 𝑋, 𝑟 ∈ ℝ* ↦ {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}) = (𝑥 ∈ 𝑋, 𝑟 ∈ ℝ* ↦ {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟})
22	21	fmpo 8044	. . . 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 12982	. . . 4 ⊢ ℝ* ∈ V
25		xpexg 7728	. . . 4 ⊢ ((𝑋 ∈ dom PsMet ∧ ℝ* ∈ V) → (𝑋 × ℝ*) ∈ V)
26	15, 24, 25	sylancl 595	. . 3 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝑋 × ℝ*) ∈ V)
27	15	pwexd 5333	. . 3 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝒫 𝑋 ∈ V)
28		fex2 7912	. . 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 1389	. 2 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝑥 ∈ 𝑋, 𝑟 ∈ ℝ* ↦ {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}) ∈ V)
30	1, 12, 13, 29	fvmptd2 6979	1 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (ball‘𝐷) = (𝑥 ∈ 𝑋, 𝑟 ∈ ℝ* ↦ {𝑦 ∈ 𝑋 ∣ (𝑥𝐷𝑦) < 𝑟}))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∀wral 3075  {crab 3413  Vcvv 3453   ⊆ wss 3902  𝒫 cpw 4552   class class class wbr 5097   × cxp 5641  dom cdm 5643  ⟶wf 6512  ‘cfv 6516  (class class class)co 7391   ∈ cmpo 7393  ℝ^*cxr 11209   < clt 11210  PsMetcpsmet 21396  ballcbl 21399

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7713  ax-cnex 11123  ax-resscn 11124

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-fv 6524  df-ov 7394  df-oprab 7395  df-mpo 7396  df-1st 7965  df-2nd 7966  df-map 8804  df-xr 11214  df-psmet 21404  df-bl 21407

This theorem is referenced by:  blfval  24432  blvalps  24433  blfps  24454